I am done

rl/Lib/site-packages/sympy/ntheory/__init__.py

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+"""
+Number theory module (primes, etc)
+"""
+
+from .generate import nextprime, prevprime, prime, primepi, primerange, \
+    randprime, Sieve, sieve, primorial, cycle_length, composite, compositepi
+from .primetest import isprime, is_gaussian_prime, is_mersenne_prime
+from .factor_ import divisors, proper_divisors, factorint, multiplicity, \
+    multiplicity_in_factorial, perfect_power, pollard_pm1, pollard_rho, \
+    primefactors, totient, \
+    divisor_count, proper_divisor_count, divisor_sigma, factorrat, \
+    reduced_totient, primenu, primeomega, mersenne_prime_exponent, \
+    is_perfect, is_abundant, is_deficient, is_amicable, is_carmichael, \
+    abundance, dra, drm
+
+from .partitions_ import npartitions
+from .residue_ntheory import is_primitive_root, is_quad_residue, \
+    legendre_symbol, jacobi_symbol, n_order, sqrt_mod, quadratic_residues, \
+    primitive_root, nthroot_mod, is_nthpow_residue, sqrt_mod_iter, mobius, \
+    discrete_log, quadratic_congruence, polynomial_congruence
+from .multinomial import binomial_coefficients, binomial_coefficients_list, \
+    multinomial_coefficients
+from .continued_fraction import continued_fraction_periodic, \
+    continued_fraction_iterator, continued_fraction_reduce, \
+    continued_fraction_convergents, continued_fraction
+from .digits import count_digits, digits, is_palindromic
+from .egyptian_fraction import egyptian_fraction
+from .ecm import ecm
+from .qs import qs
+__all__ = [
+    'nextprime', 'prevprime', 'prime', 'primepi', 'primerange', 'randprime',
+    'Sieve', 'sieve', 'primorial', 'cycle_length', 'composite', 'compositepi',
+
+    'isprime', 'is_gaussian_prime', 'is_mersenne_prime',
+
+
+    'divisors', 'proper_divisors', 'factorint', 'multiplicity', 'perfect_power',
+    'pollard_pm1', 'pollard_rho', 'primefactors', 'totient',
+    'divisor_count', 'proper_divisor_count', 'divisor_sigma', 'factorrat',
+    'reduced_totient', 'primenu', 'primeomega', 'mersenne_prime_exponent',
+    'is_perfect', 'is_abundant', 'is_deficient', 'is_amicable',
+    'is_carmichael', 'abundance', 'dra', 'drm', 'multiplicity_in_factorial',
+
+    'npartitions',
+
+    'is_primitive_root', 'is_quad_residue', 'legendre_symbol',
+    'jacobi_symbol', 'n_order', 'sqrt_mod', 'quadratic_residues',
+    'primitive_root', 'nthroot_mod', 'is_nthpow_residue', 'sqrt_mod_iter',
+    'mobius', 'discrete_log', 'quadratic_congruence', 'polynomial_congruence',
+
+    'binomial_coefficients', 'binomial_coefficients_list',
+    'multinomial_coefficients',
+
+    'continued_fraction_periodic', 'continued_fraction_iterator',
+    'continued_fraction_reduce', 'continued_fraction_convergents',
+    'continued_fraction',
+
+    'digits',
+    'count_digits',
+    'is_palindromic',
+
+    'egyptian_fraction',
+
+    'ecm',
+
+    'qs',
+]

rl/Lib/site-packages/sympy/ntheory/bbp_pi.py

@@ -0,0 +1,190 @@
+'''
+This implementation is a heavily modified fixed point implementation of
+BBP_formula for calculating the nth position of pi. The original hosted
+at: https://web.archive.org/web/20151116045029/http://en.literateprograms.org/Pi_with_the_BBP_formula_(Python)
+
+# Permission is hereby granted, free of charge, to any person obtaining
+# a copy of this software and associated documentation files (the
+# "Software"), to deal in the Software without restriction, including
+# without limitation the rights to use, copy, modify, merge, publish,
+# distribute, sub-license, and/or sell copies of the Software, and to
+# permit persons to whom the Software is furnished to do so, subject to
+# the following conditions:
+#
+# The above copyright notice and this permission notice shall be
+# included in all copies or substantial portions of the Software.
+#
+# THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND,
+# EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF
+# MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT.
+# IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY
+# CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT,
+# TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE
+# SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE.
+
+Modifications:
+
+1.Once the nth digit and desired number of digits is selected, the
+number of digits of working precision is calculated to ensure that
+the hexadecimal digits returned are accurate. This is calculated as
+
+    int(math.log(start + prec)/math.log(16) + prec + 3)
+    ---------------------------------------   --------
+                      /                          /
+    number of hex digits           additional digits
+
+This was checked by the following code which completed without
+errors (and dig are the digits included in the test_bbp.py file):
+
+    for i in range(0,1000):
+     for j in range(1,1000):
+      a, b = pi_hex_digits(i, j), dig[i:i+j]
+      if a != b:
+        print('%s\n%s'%(a,b))
+
+Deceasing the additional digits by 1 generated errors, so '3' is
+the smallest additional precision needed to calculate the above
+loop without errors. The following trailing 10 digits were also
+checked to be accurate (and the times were slightly faster with
+some of the constant modifications that were made):
+
+    >> from time import time
+    >> t=time();pi_hex_digits(10**2-10 + 1, 10), time()-t
+    ('e90c6cc0ac', 0.0)
+    >> t=time();pi_hex_digits(10**4-10 + 1, 10), time()-t
+    ('26aab49ec6', 0.17100000381469727)
+    >> t=time();pi_hex_digits(10**5-10 + 1, 10), time()-t
+    ('a22673c1a5', 4.7109999656677246)
+    >> t=time();pi_hex_digits(10**6-10 + 1, 10), time()-t
+    ('9ffd342362', 59.985999822616577)
+    >> t=time();pi_hex_digits(10**7-10 + 1, 10), time()-t
+    ('c1a42e06a1', 689.51800012588501)
+
+2. The while loop to evaluate whether the series has converged quits
+when the addition amount `dt` has dropped to zero.
+
+3. the formatting string to convert the decimal to hexadecimal is
+calculated for the given precision.
+
+4. pi_hex_digits(n) changed to have coefficient to the formula in an
+array (perhaps just a matter of preference).
+
+'''
+
+from sympy.utilities.misc import as_int
+
+
+def _series(j, n, prec=14):
+
+    # Left sum from the bbp algorithm
+    s = 0
+    D = _dn(n, prec)
+    D4 = 4 * D
+    d = j
+    for k in range(n + 1):
+        s += (pow(16, n - k, d) << D4) // d
+        d += 8
+
+    # Right sum iterates to infinity for full precision, but we
+    # stop at the point where one iteration is beyond the precision
+    # specified.
+
+    t = 0
+    k = n + 1
+    e = D4 - 4 # 4*(D + n - k)
+    d = 8 * k + j
+    while True:
+        dt = (1 << e) // d
+        if not dt:
+            break
+        t += dt
+        # k += 1
+        e -= 4
+        d += 8
+    total = s + t
+
+    return total
+
+
+def pi_hex_digits(n, prec=14):
+    """Returns a string containing ``prec`` (default 14) digits
+    starting at the nth digit of pi in hex. Counting of digits
+    starts at 0 and the decimal is not counted, so for n = 0 the
+    returned value starts with 3; n = 1 corresponds to the first
+    digit past the decimal point (which in hex is 2).
+
+    Parameters
+    ==========
+
+    n : non-negative integer
+    prec : non-negative integer. default = 14
+
+    Returns
+    =======
+
+    str : Returns a string containing ``prec`` digits
+          starting at the nth digit of pi in hex.
+          If ``prec`` = 0, returns empty string.
+
+    Raises
+    ======
+
+    ValueError
+        If ``n`` < 0 or ``prec`` < 0.
+        Or ``n`` or ``prec`` is not an integer.
+
+    Examples
+    ========
+
+    >>> from sympy.ntheory.bbp_pi import pi_hex_digits
+    >>> pi_hex_digits(0)
+    '3243f6a8885a30'
+    >>> pi_hex_digits(0, 3)
+    '324'
+
+    These are consistent with the following results
+
+    >>> import math
+    >>> hex(int(math.pi * 2**((14-1)*4)))
+    '0x3243f6a8885a30'
+    >>> hex(int(math.pi * 2**((3-1)*4)))
+    '0x324'
+
+    References
+    ==========
+
+    .. [1] http://www.numberworld.org/digits/Pi/
+    """
+    n, prec = as_int(n), as_int(prec)
+    if n < 0:
+        raise ValueError('n cannot be negative')
+    if prec < 0:
+        raise ValueError('prec cannot be negative')
+    if prec == 0:
+        return ''
+
+    # main of implementation arrays holding formulae coefficients
+    n -= 1
+    a = [4, 2, 1, 1]
+    j = [1, 4, 5, 6]
+
+    #formulae
+    D = _dn(n, prec)
+    x = + (a[0]*_series(j[0], n, prec)
+         - a[1]*_series(j[1], n, prec)
+         - a[2]*_series(j[2], n, prec)
+         - a[3]*_series(j[3], n, prec)) & (16**D - 1)
+
+    s = ("%0" + "%ix" % prec) % (x // 16**(D - prec))
+    return s
+
+
+def _dn(n, prec):
+    # controller for n dependence on precision
+    # n = starting digit index
+    # prec = the number of total digits to compute
+    n += 1  # because we subtract 1 for _series
+
+    # assert int(math.log(n + prec)/math.log(16)) ==\
+    #  ((n + prec).bit_length() - 1) // 4
+    return ((n + prec).bit_length() - 1) // 4 + prec + 3

rl/Lib/site-packages/sympy/ntheory/continued_fraction.py

@@ -0,0 +1,369 @@
+from __future__ import annotations
+import itertools
+from sympy.core.exprtools import factor_terms
+from sympy.core.numbers import Integer, Rational
+from sympy.core.singleton import S
+from sympy.core.symbol import Dummy
+from sympy.core.sympify import _sympify
+from sympy.utilities.misc import as_int
+
+
+def continued_fraction(a) -> list:
+    """Return the continued fraction representation of a Rational or
+    quadratic irrational.
+
+    Examples
+    ========
+
+    >>> from sympy.ntheory.continued_fraction import continued_fraction
+    >>> from sympy import sqrt
+    >>> continued_fraction((1 + 2*sqrt(3))/5)
+    [0, 1, [8, 3, 34, 3]]
+
+    See Also
+    ========
+    continued_fraction_periodic, continued_fraction_reduce, continued_fraction_convergents
+    """
+    e = _sympify(a)
+    if all(i.is_Rational for i in e.atoms()):
+        if e.is_Integer:
+            return continued_fraction_periodic(e, 1, 0)
+        elif e.is_Rational:
+            return continued_fraction_periodic(e.p, e.q, 0)
+        elif e.is_Pow and e.exp is S.Half and e.base.is_Integer:
+            return continued_fraction_periodic(0, 1, e.base)
+        elif e.is_Mul and len(e.args) == 2 and (
+                e.args[0].is_Rational and
+                e.args[1].is_Pow and
+                e.args[1].base.is_Integer and
+                e.args[1].exp is S.Half):
+            a, b = e.args
+            return continued_fraction_periodic(0, a.q, b.base, a.p)
+        else:
+            # this should not have to work very hard- no
+            # simplification, cancel, etc... which should be
+            # done by the user.  e.g. This is a fancy 1 but
+            # the user should simplify it first:
+            # sqrt(2)*(1 + sqrt(2))/(sqrt(2) + 2)
+            p, d = e.expand().as_numer_denom()
+            if d.is_Integer:
+                if p.is_Rational:
+                    return continued_fraction_periodic(p, d)
+                # look for a + b*c
+                # with c = sqrt(s)
+                if p.is_Add and len(p.args) == 2:
+                    a, bc = p.args
+                else:
+                    a = S.Zero
+                    bc = p
+                if a.is_Integer:
+                    b = S.NaN
+                    if bc.is_Mul and len(bc.args) == 2:
+                        b, c = bc.args
+                    elif bc.is_Pow:
+                        b = Integer(1)
+                        c = bc
+                    if b.is_Integer and (
+                            c.is_Pow and c.exp is S.Half and
+                            c.base.is_Integer):
+                        # (a + b*sqrt(c))/d
+                        c = c.base
+                        return continued_fraction_periodic(a, d, c, b)
+    raise ValueError(
+        'expecting a rational or quadratic irrational, not %s' % e)
+
+
+def continued_fraction_periodic(p, q, d=0, s=1) -> list:
+    r"""
+    Find the periodic continued fraction expansion of a quadratic irrational.
+
+    Compute the continued fraction expansion of a rational or a
+    quadratic irrational number, i.e. `\frac{p + s\sqrt{d}}{q}`, where
+    `p`, `q \ne 0` and `d \ge 0` are integers.
+
+    Returns the continued fraction representation (canonical form) as
+    a list of integers, optionally ending (for quadratic irrationals)
+    with list of integers representing the repeating digits.
+
+    Parameters
+    ==========
+
+    p : int
+        the rational part of the number's numerator
+    q : int
+        the denominator of the number
+    d : int, optional
+        the irrational part (discriminator) of the number's numerator
+    s : int, optional
+        the coefficient of the irrational part
+
+    Examples
+    ========
+
+    >>> from sympy.ntheory.continued_fraction import continued_fraction_periodic
+    >>> continued_fraction_periodic(3, 2, 7)
+    [2, [1, 4, 1, 1]]
+
+    Golden ratio has the simplest continued fraction expansion:
+
+    >>> continued_fraction_periodic(1, 2, 5)
+    [[1]]
+
+    If the discriminator is zero or a perfect square then the number will be a
+    rational number:
+
+    >>> continued_fraction_periodic(4, 3, 0)
+    [1, 3]
+    >>> continued_fraction_periodic(4, 3, 49)
+    [3, 1, 2]
+
+    See Also
+    ========
+
+    continued_fraction_iterator, continued_fraction_reduce
+
+    References
+    ==========
+
+    .. [1] https://en.wikipedia.org/wiki/Periodic_continued_fraction
+    .. [2] K. Rosen. Elementary Number theory and its applications.
+           Addison-Wesley, 3 Sub edition, pages 379-381, January 1992.
+
+    """
+    from sympy.functions import sqrt, floor
+
+    p, q, d, s = list(map(as_int, [p, q, d, s]))
+
+    if d < 0:
+        raise ValueError("expected non-negative for `d` but got %s" % d)
+
+    if q == 0:
+        raise ValueError("The denominator cannot be 0.")
+
+    if not s:
+        d = 0
+
+    # check for rational case
+    sd = sqrt(d)
+    if sd.is_Integer:
+        return list(continued_fraction_iterator(Rational(p + s*sd, q)))
+
+    # irrational case with sd != Integer
+    if q < 0:
+        p, q, s = -p, -q, -s
+
+    n = (p + s*sd)/q
+    if n < 0:
+        w = floor(-n)
+        f = -n - w
+        one_f = continued_fraction(1 - f)  # 1-f < 1 so cf is [0 ... [...]]
+        one_f[0] -= w + 1
+        return one_f
+
+    d *= s**2
+    sd *= s
+
+    if (d - p**2)%q:
+        d *= q**2
+        sd *= q
+        p *= q
+        q *= q
+
+    terms: list[int] = []
+    pq = {}
+
+    while (p, q) not in pq:
+        pq[(p, q)] = len(terms)
+        terms.append((p + sd)//q)
+        p = terms[-1]*q - p
+        q = (d - p**2)//q
+
+    i = pq[(p, q)]
+    return terms[:i] + [terms[i:]]  # type: ignore
+
+
+def continued_fraction_reduce(cf):
+    """
+    Reduce a continued fraction to a rational or quadratic irrational.
+
+    Compute the rational or quadratic irrational number from its
+    terminating or periodic continued fraction expansion.  The
+    continued fraction expansion (cf) should be supplied as a
+    terminating iterator supplying the terms of the expansion.  For
+    terminating continued fractions, this is equivalent to
+    ``list(continued_fraction_convergents(cf))[-1]``, only a little more
+    efficient.  If the expansion has a repeating part, a list of the
+    repeating terms should be returned as the last element from the
+    iterator.  This is the format returned by
+    continued_fraction_periodic.
+
+    For quadratic irrationals, returns the largest solution found,
+    which is generally the one sought, if the fraction is in canonical
+    form (all terms positive except possibly the first).
+
+    Examples
+    ========
+
+    >>> from sympy.ntheory.continued_fraction import continued_fraction_reduce
+    >>> continued_fraction_reduce([1, 2, 3, 4, 5])
+    225/157
+    >>> continued_fraction_reduce([-2, 1, 9, 7, 1, 2])
+    -256/233
+    >>> continued_fraction_reduce([2, 1, 2, 1, 1, 4, 1, 1, 6, 1, 1, 8]).n(10)
+    2.718281835
+    >>> continued_fraction_reduce([1, 4, 2, [3, 1]])
+    (sqrt(21) + 287)/238
+    >>> continued_fraction_reduce([[1]])
+    (1 + sqrt(5))/2
+    >>> from sympy.ntheory.continued_fraction import continued_fraction_periodic
+    >>> continued_fraction_reduce(continued_fraction_periodic(8, 5, 13))
+    (sqrt(13) + 8)/5
+
+    See Also
+    ========
+
+    continued_fraction_periodic
+
+    """
+    from sympy.solvers import solve
+
+    period = []
+    x = Dummy('x')
+
+    def untillist(cf):
+        for nxt in cf:
+            if isinstance(nxt, list):
+                period.extend(nxt)
+                yield x
+                break
+            yield nxt
+
+    a = S.Zero
+    for a in continued_fraction_convergents(untillist(cf)):
+        pass
+
+    if period:
+        y = Dummy('y')
+        solns = solve(continued_fraction_reduce(period + [y]) - y, y)
+        solns.sort()
+        pure = solns[-1]
+        rv = a.subs(x, pure).radsimp()
+    else:
+        rv = a
+    if rv.is_Add:
+        rv = factor_terms(rv)
+        if rv.is_Mul and rv.args[0] == -1:
+            rv = rv.func(*rv.args)
+    return rv
+
+
+def continued_fraction_iterator(x):
+    """
+    Return continued fraction expansion of x as iterator.
+
+    Examples
+    ========
+
+    >>> from sympy import Rational, pi
+    >>> from sympy.ntheory.continued_fraction import continued_fraction_iterator
+
+    >>> list(continued_fraction_iterator(Rational(3, 8)))
+    [0, 2, 1, 2]
+    >>> list(continued_fraction_iterator(Rational(-3, 8)))
+    [-1, 1, 1, 1, 2]
+
+    >>> for i, v in enumerate(continued_fraction_iterator(pi)):
+    ...     if i > 7:
+    ...         break
+    ...     print(v)
+    3
+    7
+    15
+    1
+    292
+    1
+    1
+    1
+
+    References
+    ==========
+
+    .. [1] https://en.wikipedia.org/wiki/Continued_fraction
+
+    """
+    from sympy.functions import floor
+    while True:
+        i = floor(x)
+        yield i
+        x -= i
+        if not x:
+            break
+        x = 1/x
+
+
+def continued_fraction_convergents(cf):
+    """
+    Return an iterator over the convergents of a continued fraction (cf).
+
+    The parameter should be in either of the following to forms:
+    - A list of partial quotients, possibly with the last element being a list
+    of repeating partial quotients, such as might be returned by
+    continued_fraction and continued_fraction_periodic.
+    - An iterable returning successive partial quotients of the continued
+    fraction, such as might be returned by continued_fraction_iterator.
+
+    In computing the convergents, the continued fraction need not be strictly
+    in canonical form (all integers, all but the first positive).
+    Rational and negative elements may be present in the expansion.
+
+    Examples
+    ========
+
+    >>> from sympy.core import pi
+    >>> from sympy import S
+    >>> from sympy.ntheory.continued_fraction import \
+            continued_fraction_convergents, continued_fraction_iterator
+
+    >>> list(continued_fraction_convergents([0, 2, 1, 2]))
+    [0, 1/2, 1/3, 3/8]
+
+    >>> list(continued_fraction_convergents([1, S('1/2'), -7, S('1/4')]))
+    [1, 3, 19/5, 7]
+
+    >>> it = continued_fraction_convergents(continued_fraction_iterator(pi))
+    >>> for n in range(7):
+    ...     print(next(it))
+    3
+    22/7
+    333/106
+    355/113
+    103993/33102
+    104348/33215
+    208341/66317
+
+    >>> it = continued_fraction_convergents([1, [1, 2]])  # sqrt(3)
+    >>> for n in range(7):
+    ...     print(next(it))
+    1
+    2
+    5/3
+    7/4
+    19/11
+    26/15
+    71/41
+
+    See Also
+    ========
+
+    continued_fraction_iterator, continued_fraction, continued_fraction_periodic
+
+    """
+    if isinstance(cf, list) and isinstance(cf[-1], list):
+        cf = itertools.chain(cf[:-1], itertools.cycle(cf[-1]))
+    p_2, q_2 = S.Zero, S.One
+    p_1, q_1 = S.One, S.Zero
+    for a in cf:
+        p, q = a*p_1 + p_2, a*q_1 + q_2
+        p_2, q_2 = p_1, q_1
+        p_1, q_1 = p, q
+        yield p/q

rl/Lib/site-packages/sympy/ntheory/digits.py

@@ -0,0 +1,150 @@
+from collections import defaultdict
+
+from sympy.utilities.iterables import multiset, is_palindromic as _palindromic
+from sympy.utilities.misc import as_int
+
+
+def digits(n, b=10, digits=None):
+    """
+    Return a list of the digits of ``n`` in base ``b``. The first
+    element in the list is ``b`` (or ``-b`` if ``n`` is negative).
+
+    Examples
+    ========
+
+    >>> from sympy.ntheory.digits import digits
+    >>> digits(35)
+    [10, 3, 5]
+
+    If the number is negative, the negative sign will be placed on the
+    base (which is the first element in the returned list):
+
+    >>> digits(-35)
+    [-10, 3, 5]
+
+    Bases other than 10 (and greater than 1) can be selected with ``b``:
+
+    >>> digits(27, b=2)
+    [2, 1, 1, 0, 1, 1]
+
+    Use the ``digits`` keyword if a certain number of digits is desired:
+
+    >>> digits(35, digits=4)
+    [10, 0, 0, 3, 5]
+
+    Parameters
+    ==========
+
+    n: integer
+        The number whose digits are returned.
+
+    b: integer
+        The base in which digits are computed.
+
+    digits: integer (or None for all digits)
+        The number of digits to be returned (padded with zeros, if
+        necessary).
+
+    See Also
+    ========
+    sympy.core.intfunc.num_digits, count_digits
+    """
+
+    b = as_int(b)
+    n = as_int(n)
+    if b < 2:
+        raise ValueError("b must be greater than 1")
+    else:
+        x, y = abs(n), []
+        while x >= b:
+            x, r = divmod(x, b)
+            y.append(r)
+        y.append(x)
+        y.append(-b if n < 0 else b)
+        y.reverse()
+        ndig = len(y) - 1
+        if digits is not None:
+            if ndig > digits:
+                raise ValueError(
+                    "For %s, at least %s digits are needed." % (n, ndig))
+            elif ndig < digits:
+                y[1:1] = [0]*(digits - ndig)
+        return y
+
+
+def count_digits(n, b=10):
+    """
+    Return a dictionary whose keys are the digits of ``n`` in the
+    given base, ``b``, with keys indicating the digits appearing in the
+    number and values indicating how many times that digit appeared.
+
+    Examples
+    ========
+
+    >>> from sympy.ntheory import count_digits
+
+    >>> count_digits(1111339)
+    {1: 4, 3: 2, 9: 1}
+
+    The digits returned are always represented in base-10
+    but the number itself can be entered in any format that is
+    understood by Python; the base of the number can also be
+    given if it is different than 10:
+
+    >>> n = 0xFA; n
+    250
+    >>> count_digits(_)
+    {0: 1, 2: 1, 5: 1}
+    >>> count_digits(n, 16)
+    {10: 1, 15: 1}
+
+    The default dictionary will return a 0 for any digit that did
+    not appear in the number. For example, which digits appear 7
+    times in ``77!``:
+
+    >>> from sympy import factorial
+    >>> c77 = count_digits(factorial(77))
+    >>> [i for i in range(10) if c77[i] == 7]
+    [1, 3, 7, 9]
+
+    See Also
+    ========
+    sympy.core.intfunc.num_digits, digits
+    """
+    rv = defaultdict(int, multiset(digits(n, b)).items())
+    rv.pop(b) if b in rv else rv.pop(-b)  # b or -b is there
+    return rv
+
+
+def is_palindromic(n, b=10):
+    """return True if ``n`` is the same when read from left to right
+    or right to left in the given base, ``b``.
+
+    Examples
+    ========
+
+    >>> from sympy.ntheory import is_palindromic
+
+    >>> all(is_palindromic(i) for i in (-11, 1, 22, 121))
+    True
+
+    The second argument allows you to test numbers in other
+    bases. For example, 88 is palindromic in base-10 but not
+    in base-8:
+
+    >>> is_palindromic(88, 8)
+    False
+
+    On the other hand, a number can be palindromic in base-8 but
+    not in base-10:
+
+    >>> 0o121, is_palindromic(0o121)
+    (81, False)
+
+    Or it might be palindromic in both bases:
+
+    >>> oct(121), is_palindromic(121, 8) and is_palindromic(121)
+    ('0o171', True)
+
+    """
+    return _palindromic(digits(n, b), 1)

rl/Lib/site-packages/sympy/ntheory/ecm.py

@@ -0,0 +1,345 @@
+from math import log
+
+from sympy.core.random import _randint
+from sympy.external.gmpy import gcd, invert, sqrt
+from sympy.utilities.misc import as_int
+from .generate import sieve, primerange
+from .primetest import isprime
+
+
+#----------------------------------------------------------------------------#
+#                                                                            #
+#                   Lenstra's Elliptic Curve Factorization                   #
+#                                                                            #
+#----------------------------------------------------------------------------#
+
+
+class Point:
+    """Montgomery form of Points in an elliptic curve.
+    In this form, the addition and doubling of points
+    does not need any y-coordinate information thus
+    decreasing the number of operations.
+    Using Montgomery form we try to perform point addition
+    and doubling in least amount of multiplications.
+
+    The elliptic curve used here is of the form
+    (E : b*y**2*z = x**3 + a*x**2*z + x*z**2).
+    The a_24 parameter is equal to (a + 2)/4.
+
+    References
+    ==========
+
+    .. [1] Kris Gaj, Soonhak Kwon, Patrick Baier, Paul Kohlbrenner, Hoang Le, Mohammed Khaleeluddin, Ramakrishna Bachimanchi,
+           Implementing the Elliptic Curve Method of Factoring in Reconfigurable Hardware,
+           Cryptographic Hardware and Embedded Systems - CHES 2006 (2006), pp. 119-133,
+           https://doi.org/10.1007/11894063_10
+           https://www.hyperelliptic.org/tanja/SHARCS/talks06/Gaj.pdf
+
+    """
+
+    def __init__(self, x_cord, z_cord, a_24, mod):
+        """
+        Initial parameters for the Point class.
+
+        Parameters
+        ==========
+
+        x_cord : X coordinate of the Point
+        z_cord : Z coordinate of the Point
+        a_24 : Parameter of the elliptic curve in Montgomery form
+        mod : modulus
+        """
+        self.x_cord = x_cord
+        self.z_cord = z_cord
+        self.a_24 = a_24
+        self.mod = mod
+
+    def __eq__(self, other):
+        """Two points are equal if X/Z of both points are equal
+        """
+        if self.a_24 != other.a_24 or self.mod != other.mod:
+            return False
+        return self.x_cord * other.z_cord % self.mod ==\
+            other.x_cord * self.z_cord % self.mod
+
+    def add(self, Q, diff):
+        """
+        Add two points self and Q where diff = self - Q. Moreover the assumption
+        is self.x_cord*Q.x_cord*(self.x_cord - Q.x_cord) != 0. This algorithm
+        requires 6 multiplications. Here the difference between the points
+        is already known and using this algorithm speeds up the addition
+        by reducing the number of multiplication required. Also in the
+        mont_ladder algorithm is constructed in a way so that the difference
+        between intermediate points is always equal to the initial point.
+        So, we always know what the difference between the point is.
+
+
+        Parameters
+        ==========
+
+        Q : point on the curve in Montgomery form
+        diff : self - Q
+
+        Examples
+        ========
+
+        >>> from sympy.ntheory.ecm import Point
+        >>> p1 = Point(11, 16, 7, 29)
+        >>> p2 = Point(13, 10, 7, 29)
+        >>> p3 = p2.add(p1, p1)
+        >>> p3.x_cord
+        23
+        >>> p3.z_cord
+        17
+        """
+        u = (self.x_cord - self.z_cord)*(Q.x_cord + Q.z_cord)
+        v = (self.x_cord + self.z_cord)*(Q.x_cord - Q.z_cord)
+        add, subt = u + v, u - v
+        x_cord = diff.z_cord * add * add % self.mod
+        z_cord = diff.x_cord * subt * subt % self.mod
+        return Point(x_cord, z_cord, self.a_24, self.mod)
+
+    def double(self):
+        """
+        Doubles a point in an elliptic curve in Montgomery form.
+        This algorithm requires 5 multiplications.
+
+        Examples
+        ========
+
+        >>> from sympy.ntheory.ecm import Point
+        >>> p1 = Point(11, 16, 7, 29)
+        >>> p2 = p1.double()
+        >>> p2.x_cord
+        13
+        >>> p2.z_cord
+        10
+        """
+        u = pow(self.x_cord + self.z_cord, 2, self.mod)
+        v = pow(self.x_cord - self.z_cord, 2, self.mod)
+        diff = u - v
+        x_cord = u*v % self.mod
+        z_cord = diff*(v + self.a_24*diff) % self.mod
+        return Point(x_cord, z_cord, self.a_24, self.mod)
+
+    def mont_ladder(self, k):
+        """
+        Scalar multiplication of a point in Montgomery form
+        using Montgomery Ladder Algorithm.
+        A total of 11 multiplications are required in each step of this
+        algorithm.
+
+        Parameters
+        ==========
+
+        k : The positive integer multiplier
+
+        Examples
+        ========
+
+        >>> from sympy.ntheory.ecm import Point
+        >>> p1 = Point(11, 16, 7, 29)
+        >>> p3 = p1.mont_ladder(3)
+        >>> p3.x_cord
+        23
+        >>> p3.z_cord
+        17
+        """
+        Q = self
+        R = self.double()
+        for i in bin(k)[3:]:
+            if  i  == '1':
+                Q = R.add(Q, self)
+                R = R.double()
+            else:
+                R = Q.add(R, self)
+                Q = Q.double()
+        return Q
+
+
+def _ecm_one_factor(n, B1=10000, B2=100000, max_curve=200, seed=None):
+    """Returns one factor of n using
+    Lenstra's 2 Stage Elliptic curve Factorization
+    with Suyama's Parameterization. Here Montgomery
+    arithmetic is used for fast computation of addition
+    and doubling of points in elliptic curve.
+
+    Explanation
+    ===========
+
+    This ECM method considers elliptic curves in Montgomery
+    form (E : b*y**2*z = x**3 + a*x**2*z + x*z**2) and involves
+    elliptic curve operations (mod N), where the elements in
+    Z are reduced (mod N). Since N is not a prime, E over FF(N)
+    is not really an elliptic curve but we can still do point additions
+    and doubling as if FF(N) was a field.
+
+    Stage 1 : The basic algorithm involves taking a random point (P) on an
+    elliptic curve in FF(N). The compute k*P using Montgomery ladder algorithm.
+    Let q be an unknown factor of N. Then the order of the curve E, |E(FF(q))|,
+    might be a smooth number that divides k. Then we have k = l * |E(FF(q))|
+    for some l. For any point belonging to the curve E, |E(FF(q))|*P = O,
+    hence k*P = l*|E(FF(q))|*P. Thus kP.z_cord = 0 (mod q), and the unknownn
+    factor of N (q) can be recovered by taking gcd(kP.z_cord, N).
+
+    Stage 2 : This is a continuation of Stage 1 if k*P != O. The idea utilize
+    the fact that even if kP != 0, the value of k might miss just one large
+    prime divisor of |E(FF(q))|. In this case we only need to compute the
+    scalar multiplication by p to get p*k*P = O. Here a second bound B2
+    restrict the size of possible values of p.
+
+    Parameters
+    ==========
+
+    n : Number to be Factored
+    B1 : Stage 1 Bound. Must be an even number.
+    B2 : Stage 2 Bound. Must be an even number.
+    max_curve : Maximum number of curves generated
+
+    Returns
+    =======
+
+    integer | None : ``n`` (if it is prime) else a non-trivial divisor of ``n``. ``None`` if not found
+
+    References
+    ==========
+
+    .. [1] Carl Pomerance, Richard Crandall, Prime Numbers: A Computational Perspective,
+           2nd Edition (2005), page 344, ISBN:978-0387252827
+    """
+    randint = _randint(seed)
+    if isprime(n):
+        return n
+
+    # When calculating T, if (B1 - 2*D) is negative, it cannot be calculated.
+    D = min(sqrt(B2), B1 // 2 - 1)
+    sieve.extend(D)
+    beta = [0] * D
+    S = [0] * D
+    k = 1
+    for p in primerange(2, B1 + 1):
+        k *= pow(p, int(log(B1, p)))
+
+    # Pre-calculate the prime numbers to be used in stage 2.
+    # Using the fact that the x-coordinates of point P and its
+    # inverse -P coincide, the number of primes to be checked
+    # in stage 2 can be reduced.
+    deltas_list = []
+    for r in range(B1 + 2*D, B2 + 2*D, 4*D):
+        deltas = set()
+        deltas.update((abs(q - r) - 1) // 2 for q in primerange(r - 2*D, r + 2*D))
+        # d in deltas iff r+(2d+1) and/or r-(2d+1) is prime
+        deltas_list.append(list(deltas))
+
+    for _ in range(max_curve):
+        #Suyama's Parametrization
+        sigma = randint(6, n - 1)
+        u = (sigma**2 - 5) % n
+        v = (4*sigma) % n
+        u_3 = pow(u, 3, n)
+
+        try:
+            # We use the elliptic curve y**2 = x**3 + a*x**2 + x
+            # where a = pow(v - u, 3, n)*(3*u + v)*invert(4*u_3*v, n) - 2
+            # However, we do not declare a because it is more convenient
+            # to use a24 = (a + 2)*invert(4, n) in the calculation.
+            a24 = pow(v - u, 3, n)*(3*u + v)*invert(16*u_3*v, n) % n
+        except ZeroDivisionError:
+            #If the invert(16*u_3*v, n) doesn't exist (i.e., g != 1)
+            g = gcd(2*u_3*v, n)
+            #If g = n, try another curve
+            if g == n:
+                continue
+            return g
+
+        Q = Point(u_3, pow(v, 3, n), a24, n)
+        Q = Q.mont_ladder(k)
+        g = gcd(Q.z_cord, n)
+
+        #Stage 1 factor
+        if g != 1 and g != n:
+            return g
+        #Stage 1 failure. Q.z = 0, Try another curve
+        elif g == n:
+            continue
+
+        #Stage 2 - Improved Standard Continuation
+        S[0] = Q
+        Q2 = Q.double()
+        S[1] = Q2.add(Q, Q)
+        beta[0] = (S[0].x_cord*S[0].z_cord) % n
+        beta[1] = (S[1].x_cord*S[1].z_cord) % n
+        for d in range(2, D):
+            S[d] = S[d - 1].add(Q2, S[d - 2])
+            beta[d] = (S[d].x_cord*S[d].z_cord) % n
+        # i.e., S[i] = Q.mont_ladder(2*i + 1)
+
+        g = 1
+        W = Q.mont_ladder(4*D)
+        T = Q.mont_ladder(B1 - 2*D)
+        R = Q.mont_ladder(B1 + 2*D)
+        for deltas in deltas_list:
+            # R = Q.mont_ladder(r) where r in range(B1 + 2*D, B2 + 2*D, 4*D)
+            alpha = (R.x_cord*R.z_cord) % n
+            for delta in deltas:
+                # We want to calculate
+                # f = R.x_cord * S[delta].z_cord - S[delta].x_cord * R.z_cord
+                f = (R.x_cord - S[delta].x_cord)*\
+                    (R.z_cord + S[delta].z_cord) - alpha + beta[delta]
+                g = (g*f) % n
+            T, R = R, R.add(W, T)
+        g = gcd(n, g)
+
+        #Stage 2 Factor found
+        if g != 1 and g != n:
+            return g
+
+
+def ecm(n, B1=10000, B2=100000, max_curve=200, seed=1234):
+    """Performs factorization using Lenstra's Elliptic curve method.
+
+    This function repeatedly calls ``_ecm_one_factor`` to compute the factors
+    of n. First all the small factors are taken out using trial division.
+    Then ``_ecm_one_factor`` is used to compute one factor at a time.
+
+    Parameters
+    ==========
+
+    n : Number to be Factored
+    B1 : Stage 1 Bound. Must be an even number.
+    B2 : Stage 2 Bound. Must be an even number.
+    max_curve : Maximum number of curves generated
+    seed : Initialize pseudorandom generator
+
+    Examples
+    ========
+
+    >>> from sympy.ntheory import ecm
+    >>> ecm(25645121643901801)
+    {5394769, 4753701529}
+    >>> ecm(9804659461513846513)
+    {4641991, 2112166839943}
+    """
+    n = as_int(n)
+    if B1 % 2 != 0 or B2 % 2 != 0:
+        raise ValueError("both bounds must be even")
+    _factors = set()
+    for prime in sieve.primerange(1, 100000):
+        if n % prime == 0:
+            _factors.add(prime)
+            while(n % prime == 0):
+                n //= prime
+    while(n > 1):
+        factor = _ecm_one_factor(n, B1, B2, max_curve, seed)
+        if factor is None:
+            raise ValueError("Increase the bounds")
+        _factors.add(factor)
+        n //= factor
+
+    factors = set()
+    for factor in _factors:
+        if isprime(factor):
+            factors.add(factor)
+            continue
+        factors |= ecm(factor)
+    return factors

rl/Lib/site-packages/sympy/ntheory/egyptian_fraction.py

@@ -0,0 +1,223 @@
+from sympy.core.containers import Tuple
+from sympy.core.numbers import (Integer, Rational)
+from sympy.core.singleton import S
+import sympy.polys
+
+from math import gcd
+
+
+def egyptian_fraction(r, algorithm="Greedy"):
+    """
+    Return the list of denominators of an Egyptian fraction
+    expansion [1]_ of the said rational `r`.
+
+    Parameters
+    ==========
+
+    r : Rational or (p, q)
+        a positive rational number, ``p/q``.
+    algorithm : { "Greedy", "Graham Jewett", "Takenouchi", "Golomb" }, optional
+        Denotes the algorithm to be used (the default is "Greedy").
+
+    Examples
+    ========
+
+    >>> from sympy import Rational
+    >>> from sympy.ntheory.egyptian_fraction import egyptian_fraction
+    >>> egyptian_fraction(Rational(3, 7))
+    [3, 11, 231]
+    >>> egyptian_fraction((3, 7), "Graham Jewett")
+    [7, 8, 9, 56, 57, 72, 3192]
+    >>> egyptian_fraction((3, 7), "Takenouchi")
+    [4, 7, 28]
+    >>> egyptian_fraction((3, 7), "Golomb")
+    [3, 15, 35]
+    >>> egyptian_fraction((11, 5), "Golomb")
+    [1, 2, 3, 4, 9, 234, 1118, 2580]
+
+    See Also
+    ========
+
+    sympy.core.numbers.Rational
+
+    Notes
+    =====
+
+    Currently the following algorithms are supported:
+
+    1) Greedy Algorithm
+
+       Also called the Fibonacci-Sylvester algorithm [2]_.
+       At each step, extract the largest unit fraction less
+       than the target and replace the target with the remainder.
+
+       It has some distinct properties:
+
+       a) Given `p/q` in lowest terms, generates an expansion of maximum
+          length `p`. Even as the numerators get large, the number of
+          terms is seldom more than a handful.
+
+       b) Uses minimal memory.
+
+       c) The terms can blow up (standard examples of this are 5/121 and
+          31/311).  The denominator is at most squared at each step
+          (doubly-exponential growth) and typically exhibits
+          singly-exponential growth.
+
+    2) Graham Jewett Algorithm
+
+       The algorithm suggested by the result of Graham and Jewett.
+       Note that this has a tendency to blow up: the length of the
+       resulting expansion is always ``2**(x/gcd(x, y)) - 1``.  See [3]_.
+
+    3) Takenouchi Algorithm
+
+       The algorithm suggested by Takenouchi (1921).
+       Differs from the Graham-Jewett algorithm only in the handling
+       of duplicates.  See [3]_.
+
+    4) Golomb's Algorithm
+
+       A method given by Golumb (1962), using modular arithmetic and
+       inverses.  It yields the same results as a method using continued
+       fractions proposed by Bleicher (1972).  See [4]_.
+
+    If the given rational is greater than or equal to 1, a greedy algorithm
+    of summing the harmonic sequence 1/1 + 1/2 + 1/3 + ... is used, taking
+    all the unit fractions of this sequence until adding one more would be
+    greater than the given number.  This list of denominators is prefixed
+    to the result from the requested algorithm used on the remainder.  For
+    example, if r is 8/3, using the Greedy algorithm, we get [1, 2, 3, 4,
+    5, 6, 7, 14, 420], where the beginning of the sequence, [1, 2, 3, 4, 5,
+    6, 7] is part of the harmonic sequence summing to 363/140, leaving a
+    remainder of 31/420, which yields [14, 420] by the Greedy algorithm.
+    The result of egyptian_fraction(Rational(8, 3), "Golomb") is [1, 2, 3,
+    4, 5, 6, 7, 14, 574, 2788, 6460, 11590, 33062, 113820], and so on.
+
+    References
+    ==========
+
+    .. [1] https://en.wikipedia.org/wiki/Egyptian_fraction
+    .. [2] https://en.wikipedia.org/wiki/Greedy_algorithm_for_Egyptian_fractions
+    .. [3] https://www.ics.uci.edu/~eppstein/numth/egypt/conflict.html
+    .. [4] https://web.archive.org/web/20180413004012/https://ami.ektf.hu/uploads/papers/finalpdf/AMI_42_from129to134.pdf
+
+    """
+
+    if not isinstance(r, Rational):
+        if isinstance(r, (Tuple, tuple)) and len(r) == 2:
+            r = Rational(*r)
+        else:
+            raise ValueError("Value must be a Rational or tuple of ints")
+    if r <= 0:
+        raise ValueError("Value must be positive")
+
+    # common cases that all methods agree on
+    x, y = r.as_numer_denom()
+    if y == 1 and x == 2:
+        return [Integer(i) for i in [1, 2, 3, 6]]
+    if x == y + 1:
+        return [S.One, y]
+
+    prefix, rem = egypt_harmonic(r)
+    if rem == 0:
+        return prefix
+    # work in Python ints
+    x, y = rem.p, rem.q
+    # assert x < y and gcd(x, y) = 1
+
+    if algorithm == "Greedy":
+        postfix = egypt_greedy(x, y)
+    elif algorithm == "Graham Jewett":
+        postfix = egypt_graham_jewett(x, y)
+    elif algorithm == "Takenouchi":
+        postfix = egypt_takenouchi(x, y)
+    elif algorithm == "Golomb":
+        postfix = egypt_golomb(x, y)
+    else:
+        raise ValueError("Entered invalid algorithm")
+    return prefix + [Integer(i) for i in postfix]
+
+
+def egypt_greedy(x, y):
+    # assumes gcd(x, y) == 1
+    if x == 1:
+        return [y]
+    else:
+        a = (-y) % x
+        b = y*(y//x + 1)
+        c = gcd(a, b)
+        if c > 1:
+            num, denom = a//c, b//c
+        else:
+            num, denom = a, b
+        return [y//x + 1] + egypt_greedy(num, denom)
+
+
+def egypt_graham_jewett(x, y):
+    # assumes gcd(x, y) == 1
+    l = [y] * x
+
+    # l is now a list of integers whose reciprocals sum to x/y.
+    # we shall now proceed to manipulate the elements of l without
+    # changing the reciprocated sum until all elements are unique.
+
+    while len(l) != len(set(l)):
+        l.sort()  # so the list has duplicates. find a smallest pair
+        for i in range(len(l) - 1):
+            if l[i] == l[i + 1]:
+                break
+        # we have now identified a pair of identical
+        # elements: l[i] and l[i + 1].
+        # now comes the application of the result of graham and jewett:
+        l[i + 1] = l[i] + 1
+        # and we just iterate that until the list has no duplicates.
+        l.append(l[i]*(l[i] + 1))
+    return sorted(l)
+
+
+def egypt_takenouchi(x, y):
+    # assumes gcd(x, y) == 1
+    # special cases for 3/y
+    if x == 3:
+        if y % 2 == 0:
+            return [y//2, y]
+        i = (y - 1)//2
+        j = i + 1
+        k = j + i
+        return [j, k, j*k]
+    l = [y] * x
+    while len(l) != len(set(l)):
+        l.sort()
+        for i in range(len(l) - 1):
+            if l[i] == l[i + 1]:
+                break
+        k = l[i]
+        if k % 2 == 0:
+            l[i] = l[i] // 2
+            del l[i + 1]
+        else:
+            l[i], l[i + 1] = (k + 1)//2, k*(k + 1)//2
+    return sorted(l)
+
+
+def egypt_golomb(x, y):
+    # assumes x < y and gcd(x, y) == 1
+    if x == 1:
+        return [y]
+    xp = sympy.polys.ZZ.invert(int(x), int(y))
+    rv = [xp*y]
+    rv.extend(egypt_golomb((x*xp - 1)//y, xp))
+    return sorted(rv)
+
+
+def egypt_harmonic(r):
+    # assumes r is Rational
+    rv = []
+    d = S.One
+    acc = S.Zero
+    while acc + 1/d <= r:
+        acc += 1/d
+        rv.append(d)
+        d += 1
+    return (rv, r - acc)

rl/Lib/site-packages/sympy/ntheory/elliptic_curve.py

@@ -0,0 +1,397 @@
+from sympy.core.numbers import oo
+from sympy.core.symbol import symbols
+from sympy.polys.domains import FiniteField, QQ, RationalField, FF
+from sympy.polys.polytools import Poly
+from sympy.solvers.solvers import solve
+from sympy.utilities.iterables import is_sequence
+from sympy.utilities.misc import as_int
+from .factor_ import divisors
+from .residue_ntheory import polynomial_congruence
+
+
+class EllipticCurve:
+    """
+    Create the following Elliptic Curve over domain.
+
+    `y^{2} + a_{1} x y + a_{3} y = x^{3} + a_{2} x^{2} + a_{4} x + a_{6}`
+
+    The default domain is ``QQ``. If no coefficient ``a1``, ``a2``, ``a3``,
+    is given then it creates a curve with the following form:
+
+    `y^{2} = x^{3} + a_{4} x + a_{6}`
+
+    Examples
+    ========
+
+    References
+    ==========
+
+    .. [1] J. Silverman "A Friendly Introduction to Number Theory" Third Edition
+    .. [2] https://mathworld.wolfram.com/EllipticDiscriminant.html
+    .. [3] G. Hardy, E. Wright "An Introduction to the Theory of Numbers" Sixth Edition
+
+    """
+
+    def __init__(self, a4, a6, a1=0, a2=0, a3=0, modulus=0):
+        if modulus == 0:
+            domain = QQ
+        else:
+            domain = FF(modulus)
+        a1, a2, a3, a4, a6 = map(domain.convert, (a1, a2, a3, a4, a6))
+        self._domain = domain
+        self.modulus = modulus
+        # Calculate discriminant
+        b2 = a1**2 + 4 * a2
+        b4 = 2 * a4 + a1 * a3
+        b6 = a3**2 + 4 * a6
+        b8 = a1**2 * a6 + 4 * a2 * a6 - a1 * a3 * a4 + a2 * a3**2 - a4**2
+        self._b2, self._b4, self._b6, self._b8 = b2, b4, b6, b8
+        self._discrim = -b2**2 * b8 - 8 * b4**3 - 27 * b6**2 + 9 * b2 * b4 * b6
+        self._a1 = a1
+        self._a2 = a2
+        self._a3 = a3
+        self._a4 = a4
+        self._a6 = a6
+        x, y, z = symbols('x y z')
+        self.x, self.y, self.z = x, y, z
+        self._poly = Poly(y**2*z + a1*x*y*z + a3*y*z**2 - x**3 - a2*x**2*z - a4*x*z**2 - a6*z**3, domain=domain)
+        if isinstance(self._domain, FiniteField):
+            self._rank = 0
+        elif isinstance(self._domain, RationalField):
+            self._rank = None
+
+    def __call__(self, x, y, z=1):
+        return EllipticCurvePoint(x, y, z, self)
+
+    def __contains__(self, point):
+        if is_sequence(point):
+            if len(point) == 2:
+                z1 = 1
+            else:
+                z1 = point[2]
+            x1, y1 = point[:2]
+        elif isinstance(point, EllipticCurvePoint):
+            x1, y1, z1 = point.x, point.y, point.z
+        else:
+            raise ValueError('Invalid point.')
+        if self.characteristic == 0 and z1 == 0:
+            return True
+        return self._poly.subs({self.x: x1, self.y: y1, self.z: z1}) == 0
+
+    def __repr__(self):
+        return self._poly.__repr__()
+
+    def minimal(self):
+        """
+        Return minimal Weierstrass equation.
+
+        Examples
+        ========
+
+        >>> from sympy.ntheory.elliptic_curve import EllipticCurve
+
+        >>> e1 = EllipticCurve(-10, -20, 0, -1, 1)
+        >>> e1.minimal()
+        Poly(-x**3 + 13392*x*z**2 + y**2*z + 1080432*z**3, x, y, z, domain='QQ')
+
+        """
+        char = self.characteristic
+        if char == 2:
+            return self
+        if char == 3:
+            return EllipticCurve(self._b4/2, self._b6/4, a2=self._b2/4, modulus=self.modulus)
+        c4 = self._b2**2 - 24*self._b4
+        c6 = -self._b2**3 + 36*self._b2*self._b4 - 216*self._b6
+        return EllipticCurve(-27*c4, -54*c6, modulus=self.modulus)
+
+    def points(self):
+        """
+        Return points of curve over Finite Field.
+
+        Examples
+        ========
+
+        >>> from sympy.ntheory.elliptic_curve import EllipticCurve
+        >>> e2 = EllipticCurve(1, 1, 1, 1, 1, modulus=5)
+        >>> e2.points()
+        {(0, 2), (1, 4), (2, 0), (2, 2), (3, 0), (3, 1), (4, 0)}
+
+        """
+
+        char = self.characteristic
+        all_pt = set()
+        if char >= 1:
+            for i in range(char):
+                congruence_eq = self._poly.subs({self.x: i, self.z: 1}).expr
+                sol = polynomial_congruence(congruence_eq, char)
+                all_pt.update((i, num) for num in sol)
+            return all_pt
+        else:
+            raise ValueError("Infinitely many points")
+
+    def points_x(self, x):
+        """Returns points on the curve for the given x-coordinate."""
+        pt = []
+        if self._domain == QQ:
+            for y in solve(self._poly.subs(self.x, x)):
+                pt.append((x, y))
+        else:
+            congruence_eq = self._poly.subs({self.x: x, self.z: 1}).expr
+            for y in polynomial_congruence(congruence_eq, self.characteristic):
+                pt.append((x, y))
+        return pt
+
+    def torsion_points(self):
+        """
+        Return torsion points of curve over Rational number.
+
+        Return point objects those are finite order.
+        According to Nagell-Lutz theorem, torsion point p(x, y)
+        x and y are integers, either y = 0 or y**2 is divisor
+        of discriminent. According to Mazur's theorem, there are
+        at most 15 points in torsion collection.
+
+        Examples
+        ========
+
+        >>> from sympy.ntheory.elliptic_curve import EllipticCurve
+        >>> e2 = EllipticCurve(-43, 166)
+        >>> sorted(e2.torsion_points())
+        [(-5, -16), (-5, 16), O, (3, -8), (3, 8), (11, -32), (11, 32)]
+
+        """
+        if self.characteristic > 0:
+            raise ValueError("No torsion point for Finite Field.")
+        l = [EllipticCurvePoint.point_at_infinity(self)]
+        for xx in solve(self._poly.subs({self.y: 0, self.z: 1})):
+            if xx.is_rational:
+                l.append(self(xx, 0))
+        for i in divisors(self.discriminant, generator=True):
+            j = int(i**.5)
+            if j**2 == i:
+                for xx in solve(self._poly.subs({self.y: j, self.z: 1})):
+                    if not xx.is_rational:
+                        continue
+                    p = self(xx, j)
+                    if p.order() != oo:
+                        l.extend([p, -p])
+        return l
+
+    @property
+    def characteristic(self):
+        """
+        Return domain characteristic.
+
+        Examples
+        ========
+
+        >>> from sympy.ntheory.elliptic_curve import EllipticCurve
+        >>> e2 = EllipticCurve(-43, 166)
+        >>> e2.characteristic
+        0
+
+        """
+        return self._domain.characteristic()
+
+    @property
+    def discriminant(self):
+        """
+        Return curve discriminant.
+
+        Examples
+        ========
+
+        >>> from sympy.ntheory.elliptic_curve import EllipticCurve
+        >>> e2 = EllipticCurve(0, 17)
+        >>> e2.discriminant
+        -124848
+
+        """
+        return int(self._discrim)
+
+    @property
+    def is_singular(self):
+        """
+        Return True if curve discriminant is equal to zero.
+        """
+        return self.discriminant == 0
+
+    @property
+    def j_invariant(self):
+        """
+        Return curve j-invariant.
+
+        Examples
+        ========
+
+        >>> from sympy.ntheory.elliptic_curve import EllipticCurve
+        >>> e1 = EllipticCurve(-2, 0, 0, 1, 1)
+        >>> e1.j_invariant
+        1404928/389
+
+        """
+        c4 = self._b2**2 - 24*self._b4
+        return self._domain.to_sympy(c4**3 / self._discrim)
+
+    @property
+    def order(self):
+        """
+        Number of points in Finite field.
+
+        Examples
+        ========
+
+        >>> from sympy.ntheory.elliptic_curve import EllipticCurve
+        >>> e2 = EllipticCurve(1, 0, modulus=19)
+        >>> e2.order
+        19
+
+        """
+        if self.characteristic == 0:
+            raise NotImplementedError("Still not implemented")
+        return len(self.points())
+
+    @property
+    def rank(self):
+        """
+        Number of independent points of infinite order.
+
+        For Finite field, it must be 0.
+        """
+        if self._rank is not None:
+            return self._rank
+        raise NotImplementedError("Still not implemented")
+
+
+class EllipticCurvePoint:
+    """
+    Point of Elliptic Curve
+
+    Examples
+    ========
+
+    >>> from sympy.ntheory.elliptic_curve import EllipticCurve
+    >>> e1 = EllipticCurve(-17, 16)
+    >>> p1 = e1(0, -4, 1)
+    >>> p2 = e1(1, 0)
+    >>> p1 + p2
+    (15, -56)
+    >>> e3 = EllipticCurve(-1, 9)
+    >>> e3(1, -3) * 3
+    (664/169, 17811/2197)
+    >>> (e3(1, -3) * 3).order()
+    oo
+    >>> e2 = EllipticCurve(-2, 0, 0, 1, 1)
+    >>> p = e2(-1,1)
+    >>> q = e2(0, -1)
+    >>> p+q
+    (4, 8)
+    >>> p-q
+    (1, 0)
+    >>> 3*p-5*q
+    (328/361, -2800/6859)
+    """
+
+    @staticmethod
+    def point_at_infinity(curve):
+        return EllipticCurvePoint(0, 1, 0, curve)
+
+    def __init__(self, x, y, z, curve):
+        dom = curve._domain.convert
+        self.x = dom(x)
+        self.y = dom(y)
+        self.z = dom(z)
+        self._curve = curve
+        self._domain = self._curve._domain
+        if not self._curve.__contains__(self):
+            raise ValueError("The curve does not contain this point")
+
+    def __add__(self, p):
+        if self.z == 0:
+            return p
+        if p.z == 0:
+            return self
+        x1, y1 = self.x/self.z, self.y/self.z
+        x2, y2 = p.x/p.z, p.y/p.z
+        a1 = self._curve._a1
+        a2 = self._curve._a2
+        a3 = self._curve._a3
+        a4 = self._curve._a4
+        a6 = self._curve._a6
+        if x1 != x2:
+            slope = (y1 - y2) / (x1 - x2)
+            yint = (y1 * x2 - y2 * x1) / (x2 - x1)
+        else:
+            if (y1 + y2) == 0:
+                return self.point_at_infinity(self._curve)
+            slope = (3 * x1**2 + 2*a2*x1 + a4 - a1*y1) / (a1 * x1 + a3 + 2 * y1)
+            yint = (-x1**3 + a4*x1 + 2*a6 - a3*y1) / (a1*x1 + a3 + 2*y1)
+        x3 = slope**2 + a1*slope - a2 - x1 - x2
+        y3 = -(slope + a1) * x3 - yint - a3
+        return self._curve(x3, y3, 1)
+
+    def __lt__(self, other):
+        return (self.x, self.y, self.z) < (other.x, other.y, other.z)
+
+    def __mul__(self, n):
+        n = as_int(n)
+        r = self.point_at_infinity(self._curve)
+        if n == 0:
+            return r
+        if n < 0:
+            return -self * -n
+        p = self
+        while n:
+            if n & 1:
+                r = r + p
+            n >>= 1
+            p = p + p
+        return r
+
+    def __rmul__(self, n):
+        return self * n
+
+    def __neg__(self):
+        return EllipticCurvePoint(self.x, -self.y - self._curve._a1*self.x - self._curve._a3, self.z, self._curve)
+
+    def __repr__(self):
+        if self.z == 0:
+            return 'O'
+        dom = self._curve._domain
+        try:
+            return '({}, {})'.format(dom.to_sympy(self.x), dom.to_sympy(self.y))
+        except TypeError:
+            pass
+        return '({}, {})'.format(self.x, self.y)
+
+    def __sub__(self, other):
+        return self.__add__(-other)
+
+    def order(self):
+        """
+        Return point order n where nP = 0.
+
+        """
+        if self.z == 0:
+            return 1
+        if self.y == 0:  # P = -P
+            return 2
+        p = self * 2
+        if p.y == -self.y:  # 2P = -P
+            return 3
+        i = 2
+        if self._domain != QQ:
+            while int(p.x) == p.x and int(p.y) == p.y:
+                p = self + p
+                i += 1
+                if p.z == 0:
+                    return i
+            return oo
+        while p.x.numerator == p.x and p.y.numerator == p.y:
+            p = self + p
+            i += 1
+            if i > 12:
+                return oo
+            if p.z == 0:
+                return i
+        return oo

rl/Lib/site-packages/sympy/ntheory/modular.py

@@ -0,0 +1,291 @@
+from math import prod
+
+from sympy.external.gmpy import gcd, gcdext
+from sympy.ntheory.primetest import isprime
+from sympy.polys.domains import ZZ
+from sympy.polys.galoistools import gf_crt, gf_crt1, gf_crt2
+from sympy.utilities.misc import as_int
+
+
+def symmetric_residue(a, m):
+    """Return the residual mod m such that it is within half of the modulus.
+
+    >>> from sympy.ntheory.modular import symmetric_residue
+    >>> symmetric_residue(1, 6)
+    1
+    >>> symmetric_residue(4, 6)
+    -2
+    """
+    if a <= m // 2:
+        return a
+    return a - m
+
+
+def crt(m, v, symmetric=False, check=True):
+    r"""Chinese Remainder Theorem.
+
+    The moduli in m are assumed to be pairwise coprime.  The output
+    is then an integer f, such that f = v_i mod m_i for each pair out
+    of v and m. If ``symmetric`` is False a positive integer will be
+    returned, else \|f\| will be less than or equal to the LCM of the
+    moduli, and thus f may be negative.
+
+    If the moduli are not co-prime the correct result will be returned
+    if/when the test of the result is found to be incorrect. This result
+    will be None if there is no solution.
+
+    The keyword ``check`` can be set to False if it is known that the moduli
+    are coprime.
+
+    Examples
+    ========
+
+    As an example consider a set of residues ``U = [49, 76, 65]``
+    and a set of moduli ``M = [99, 97, 95]``. Then we have::
+
+       >>> from sympy.ntheory.modular import crt
+
+       >>> crt([99, 97, 95], [49, 76, 65])
+       (639985, 912285)
+
+    This is the correct result because::
+
+       >>> [639985 % m for m in [99, 97, 95]]
+       [49, 76, 65]
+
+    If the moduli are not co-prime, you may receive an incorrect result
+    if you use ``check=False``:
+
+       >>> crt([12, 6, 17], [3, 4, 2], check=False)
+       (954, 1224)
+       >>> [954 % m for m in [12, 6, 17]]
+       [6, 0, 2]
+       >>> crt([12, 6, 17], [3, 4, 2]) is None
+       True
+       >>> crt([3, 6], [2, 5])
+       (5, 6)
+
+    Note: the order of gf_crt's arguments is reversed relative to crt,
+    and that solve_congruence takes residue, modulus pairs.
+
+    Programmer's note: rather than checking that all pairs of moduli share
+    no GCD (an O(n**2) test) and rather than factoring all moduli and seeing
+    that there is no factor in common, a check that the result gives the
+    indicated residuals is performed -- an O(n) operation.
+
+    See Also
+    ========
+
+    solve_congruence
+    sympy.polys.galoistools.gf_crt : low level crt routine used by this routine
+    """
+    if check:
+        m = list(map(as_int, m))
+        v = list(map(as_int, v))
+
+    result = gf_crt(v, m, ZZ)
+    mm = prod(m)
+
+    if check:
+        if not all(v % m == result % m for v, m in zip(v, m)):
+            result = solve_congruence(*list(zip(v, m)),
+                    check=False, symmetric=symmetric)
+            if result is None:
+                return result
+            result, mm = result
+
+    if symmetric:
+        return int(symmetric_residue(result, mm)), int(mm)
+    return int(result), int(mm)
+
+
+def crt1(m):
+    """First part of Chinese Remainder Theorem, for multiple application.
+
+    Examples
+    ========
+
+    >>> from sympy.ntheory.modular import crt, crt1, crt2
+    >>> m = [99, 97, 95]
+    >>> v = [49, 76, 65]
+
+    The following two codes have the same result.
+
+    >>> crt(m, v)
+    (639985, 912285)
+
+    >>> mm, e, s = crt1(m)
+    >>> crt2(m, v, mm, e, s)
+    (639985, 912285)
+
+    However, it is faster when we want to fix ``m`` and
+    compute for multiple ``v``, i.e. the following cases:
+
+    >>> mm, e, s = crt1(m)
+    >>> vs = [[52, 21, 37], [19, 46, 76]]
+    >>> for v in vs:
+    ...     print(crt2(m, v, mm, e, s))
+    (397042, 912285)
+    (803206, 912285)
+
+    See Also
+    ========
+
+    sympy.polys.galoistools.gf_crt1 : low level crt routine used by this routine
+    sympy.ntheory.modular.crt
+    sympy.ntheory.modular.crt2
+
+    """
+
+    return gf_crt1(m, ZZ)
+
+
+def crt2(m, v, mm, e, s, symmetric=False):
+    """Second part of Chinese Remainder Theorem, for multiple application.
+
+    See ``crt1`` for usage.
+
+    Examples
+    ========
+
+    >>> from sympy.ntheory.modular import crt1, crt2
+    >>> mm, e, s = crt1([18, 42, 6])
+    >>> crt2([18, 42, 6], [0, 0, 0], mm, e, s)
+    (0, 4536)
+
+    See Also
+    ========
+
+    sympy.polys.galoistools.gf_crt2 : low level crt routine used by this routine
+    sympy.ntheory.modular.crt
+    sympy.ntheory.modular.crt1
+
+    """
+
+    result = gf_crt2(v, m, mm, e, s, ZZ)
+
+    if symmetric:
+        return int(symmetric_residue(result, mm)), int(mm)
+    return int(result), int(mm)
+
+
+def solve_congruence(*remainder_modulus_pairs, **hint):
+    """Compute the integer ``n`` that has the residual ``ai`` when it is
+    divided by ``mi`` where the ``ai`` and ``mi`` are given as pairs to
+    this function: ((a1, m1), (a2, m2), ...). If there is no solution,
+    return None. Otherwise return ``n`` and its modulus.
+
+    The ``mi`` values need not be co-prime. If it is known that the moduli are
+    not co-prime then the hint ``check`` can be set to False (default=True) and
+    the check for a quicker solution via crt() (valid when the moduli are
+    co-prime) will be skipped.
+
+    If the hint ``symmetric`` is True (default is False), the value of ``n``
+    will be within 1/2 of the modulus, possibly negative.
+
+    Examples
+    ========
+
+    >>> from sympy.ntheory.modular import solve_congruence
+
+    What number is 2 mod 3, 3 mod 5 and 2 mod 7?
+
+    >>> solve_congruence((2, 3), (3, 5), (2, 7))
+    (23, 105)
+    >>> [23 % m for m in [3, 5, 7]]
+    [2, 3, 2]
+
+    If you prefer to work with all remainder in one list and
+    all moduli in another, send the arguments like this:
+
+    >>> solve_congruence(*zip((2, 3, 2), (3, 5, 7)))
+    (23, 105)
+
+    The moduli need not be co-prime; in this case there may or
+    may not be a solution:
+
+    >>> solve_congruence((2, 3), (4, 6)) is None
+    True
+
+    >>> solve_congruence((2, 3), (5, 6))
+    (5, 6)
+
+    The symmetric flag will make the result be within 1/2 of the modulus:
+
+    >>> solve_congruence((2, 3), (5, 6), symmetric=True)
+    (-1, 6)
+
+    See Also
+    ========
+
+    crt : high level routine implementing the Chinese Remainder Theorem
+
+    """
+    def combine(c1, c2):
+        """Return the tuple (a, m) which satisfies the requirement
+        that n = a + i*m satisfy n = a1 + j*m1 and n = a2 = k*m2.
+
+        References
+        ==========
+
+        .. [1] https://en.wikipedia.org/wiki/Method_of_successive_substitution
+        """
+        a1, m1 = c1
+        a2, m2 = c2
+        a, b, c = m1, a2 - a1, m2
+        g = gcd(a, b, c)
+        a, b, c = [i//g for i in [a, b, c]]
+        if a != 1:
+            g, inv_a, _ = gcdext(a, c)
+            if g != 1:
+                return None
+            b *= inv_a
+        a, m = a1 + m1*b, m1*c
+        return a, m
+
+    rm = remainder_modulus_pairs
+    symmetric = hint.get('symmetric', False)
+
+    if hint.get('check', True):
+        rm = [(as_int(r), as_int(m)) for r, m in rm]
+
+        # ignore redundant pairs but raise an error otherwise; also
+        # make sure that a unique set of bases is sent to gf_crt if
+        # they are all prime.
+        #
+        # The routine will work out less-trivial violations and
+        # return None, e.g. for the pairs (1,3) and (14,42) there
+        # is no answer because 14 mod 42 (having a gcd of 14) implies
+        # (14/2) mod (42/2), (14/7) mod (42/7) and (14/14) mod (42/14)
+        # which, being 0 mod 3, is inconsistent with 1 mod 3. But to
+        # preprocess the input beyond checking of another pair with 42
+        # or 3 as the modulus (for this example) is not necessary.
+        uniq = {}
+        for r, m in rm:
+            r %= m
+            if m in uniq:
+                if r != uniq[m]:
+                    return None
+                continue
+            uniq[m] = r
+        rm = [(r, m) for m, r in uniq.items()]
+        del uniq
+
+        # if the moduli are co-prime, the crt will be significantly faster;
+        # checking all pairs for being co-prime gets to be slow but a prime
+        # test is a good trade-off
+        if all(isprime(m) for r, m in rm):
+            r, m = list(zip(*rm))
+            return crt(m, r, symmetric=symmetric, check=False)
+
+    rv = (0, 1)
+    for rmi in rm:
+        rv = combine(rv, rmi)
+        if rv is None:
+            break
+        n, m = rv
+        n = n % m
+    else:
+        if symmetric:
+            return symmetric_residue(n, m), m
+        return n, m

rl/Lib/site-packages/sympy/ntheory/multinomial.py

@@ -0,0 +1,188 @@
+from sympy.utilities.misc import as_int
+
+
+def binomial_coefficients(n):
+    """Return a dictionary containing pairs :math:`{(k1,k2) : C_kn}` where
+    :math:`C_kn` are binomial coefficients and :math:`n=k1+k2`.
+
+    Examples
+    ========
+
+    >>> from sympy.ntheory import binomial_coefficients
+    >>> binomial_coefficients(9)
+    {(0, 9): 1, (1, 8): 9, (2, 7): 36, (3, 6): 84,
+     (4, 5): 126, (5, 4): 126, (6, 3): 84, (7, 2): 36, (8, 1): 9, (9, 0): 1}
+
+    See Also
+    ========
+
+    binomial_coefficients_list, multinomial_coefficients
+    """
+    n = as_int(n)
+    d = {(0, n): 1, (n, 0): 1}
+    a = 1
+    for k in range(1, n//2 + 1):
+        a = (a * (n - k + 1))//k
+        d[k, n - k] = d[n - k, k] = a
+    return d
+
+
+def binomial_coefficients_list(n):
+    """ Return a list of binomial coefficients as rows of the Pascal's
+    triangle.
+
+    Examples
+    ========
+
+    >>> from sympy.ntheory import binomial_coefficients_list
+    >>> binomial_coefficients_list(9)
+    [1, 9, 36, 84, 126, 126, 84, 36, 9, 1]
+
+    See Also
+    ========
+
+    binomial_coefficients, multinomial_coefficients
+    """
+    n = as_int(n)
+    d = [1] * (n + 1)
+    a = 1
+    for k in range(1, n//2 + 1):
+        a = (a * (n - k + 1))//k
+        d[k] = d[n - k] = a
+    return d
+
+
+def multinomial_coefficients(m, n):
+    r"""Return a dictionary containing pairs ``{(k1,k2,..,km) : C_kn}``
+    where ``C_kn`` are multinomial coefficients such that
+    ``n=k1+k2+..+km``.
+
+    Examples
+    ========
+
+    >>> from sympy.ntheory import multinomial_coefficients
+    >>> multinomial_coefficients(2, 5) # indirect doctest
+    {(0, 5): 1, (1, 4): 5, (2, 3): 10, (3, 2): 10, (4, 1): 5, (5, 0): 1}
+
+    Notes
+    =====
+
+    The algorithm is based on the following result:
+
+    .. math::
+        \binom{n}{k_1, \ldots, k_m} =
+        \frac{k_1 + 1}{n - k_1} \sum_{i=2}^m \binom{n}{k_1 + 1, \ldots, k_i - 1, \ldots}
+
+    Code contributed to Sage by Yann Laigle-Chapuy, copied with permission
+    of the author.
+
+    See Also
+    ========
+
+    binomial_coefficients_list, binomial_coefficients
+    """
+    m = as_int(m)
+    n = as_int(n)
+    if not m:
+        if n:
+            return {}
+        return {(): 1}
+    if m == 2:
+        return binomial_coefficients(n)
+    if m >= 2*n and n > 1:
+        return dict(multinomial_coefficients_iterator(m, n))
+    t = [n] + [0] * (m - 1)
+    r = {tuple(t): 1}
+    if n:
+        j = 0  # j will be the leftmost nonzero position
+    else:
+        j = m
+    # enumerate tuples in co-lex order
+    while j < m - 1:
+        # compute next tuple
+        tj = t[j]
+        if j:
+            t[j] = 0
+            t[0] = tj
+        if tj > 1:
+            t[j + 1] += 1
+            j = 0
+            start = 1
+            v = 0
+        else:
+            j += 1
+            start = j + 1
+            v = r[tuple(t)]
+            t[j] += 1
+        # compute the value
+        # NB: the initialization of v was done above
+        for k in range(start, m):
+            if t[k]:
+                t[k] -= 1
+                v += r[tuple(t)]
+                t[k] += 1
+        t[0] -= 1
+        r[tuple(t)] = (v * tj) // (n - t[0])
+    return r
+
+
+def multinomial_coefficients_iterator(m, n, _tuple=tuple):
+    """multinomial coefficient iterator
+
+    This routine has been optimized for `m` large with respect to `n` by taking
+    advantage of the fact that when the monomial tuples `t` are stripped of
+    zeros, their coefficient is the same as that of the monomial tuples from
+    ``multinomial_coefficients(n, n)``. Therefore, the latter coefficients are
+    precomputed to save memory and time.
+
+    >>> from sympy.ntheory.multinomial import multinomial_coefficients
+    >>> m53, m33 = multinomial_coefficients(5,3), multinomial_coefficients(3,3)
+    >>> m53[(0,0,0,1,2)] == m53[(0,0,1,0,2)] == m53[(1,0,2,0,0)] == m33[(0,1,2)]
+    True
+
+    Examples
+    ========
+
+    >>> from sympy.ntheory.multinomial import multinomial_coefficients_iterator
+    >>> it = multinomial_coefficients_iterator(20,3)
+    >>> next(it)
+    ((3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0), 1)
+    """
+    m = as_int(m)
+    n = as_int(n)
+    if m < 2*n or n == 1:
+        mc = multinomial_coefficients(m, n)
+        yield from mc.items()
+    else:
+        mc = multinomial_coefficients(n, n)
+        mc1 = {}
+        for k, v in mc.items():
+            mc1[_tuple(filter(None, k))] = v
+        mc = mc1
+
+        t = [n] + [0] * (m - 1)
+        t1 = _tuple(t)
+        b = _tuple(filter(None, t1))
+        yield (t1, mc[b])
+        if n:
+            j = 0  # j will be the leftmost nonzero position
+        else:
+            j = m
+        # enumerate tuples in co-lex order
+        while j < m - 1:
+            # compute next tuple
+            tj = t[j]
+            if j:
+                t[j] = 0
+                t[0] = tj
+            if tj > 1:
+                t[j + 1] += 1
+                j = 0
+            else:
+                j += 1
+                t[j] += 1
+
+            t[0] -= 1
+            t1 = _tuple(t)
+            b = _tuple(filter(None, t1))
+            yield (t1, mc[b])

rl/Lib/site-packages/sympy/ntheory/partitions_.py

@@ -0,0 +1,278 @@
+from mpmath.libmp import (fzero, from_int, from_rational,
+    fone, fhalf, bitcount, to_int, mpf_mul, mpf_div, mpf_sub,
+    mpf_add, mpf_sqrt, mpf_pi, mpf_cosh_sinh, mpf_cos, mpf_sin)
+from sympy.external.gmpy import gcd, legendre, jacobi
+from .residue_ntheory import _sqrt_mod_prime_power, is_quad_residue
+from sympy.utilities.decorator import deprecated
+from sympy.utilities.memoization import recurrence_memo
+
+import math
+from itertools import count
+
+def _pre():
+    maxn = 10**5
+    global _factor
+    global _totient
+    _factor = [0]*maxn
+    _totient = [1]*maxn
+    lim = int(maxn**0.5) + 5
+    for i in range(2, lim):
+        if _factor[i] == 0:
+            for j in range(i*i, maxn, i):
+                if _factor[j] == 0:
+                    _factor[j] = i
+    for i in range(2, maxn):
+        if _factor[i] == 0:
+            _factor[i] = i
+            _totient[i] = i-1
+            continue
+        x = _factor[i]
+        y = i//x
+        if y % x == 0:
+            _totient[i] = _totient[y]*x
+        else:
+            _totient[i] = _totient[y]*(x - 1)
+
+def _a(n, k, prec):
+    """ Compute the inner sum in HRR formula [1]_
+
+    References
+    ==========
+
+    .. [1] https://msp.org/pjm/1956/6-1/pjm-v6-n1-p18-p.pdf
+
+    """
+    if k == 1:
+        return fone
+
+    k1 = k
+    e = 0
+    p = _factor[k]
+    while k1 % p == 0:
+        k1 //= p
+        e += 1
+    k2 = k//k1 # k2 = p^e
+    v = 1 - 24*n
+    pi = mpf_pi(prec)
+
+    if k1 == 1:
+        # k  = p^e
+        if p == 2:
+            mod = 8*k
+            v = mod + v % mod
+            v = (v*pow(9, k - 1, mod)) % mod
+            m = _sqrt_mod_prime_power(v, 2, e + 3)[0]
+            arg = mpf_div(mpf_mul(
+                from_int(4*m), pi, prec), from_int(mod), prec)
+            return mpf_mul(mpf_mul(
+                from_int((-1)**e*jacobi(m - 1, m)),
+                mpf_sqrt(from_int(k), prec), prec),
+                mpf_sin(arg, prec), prec)
+        if p == 3:
+            mod = 3*k
+            v = mod + v % mod
+            if e > 1:
+                v = (v*pow(64, k//3 - 1, mod)) % mod
+            m = _sqrt_mod_prime_power(v, 3, e + 1)[0]
+            arg = mpf_div(mpf_mul(from_int(4*m), pi, prec),
+                from_int(mod), prec)
+            return mpf_mul(mpf_mul(
+                from_int(2*(-1)**(e + 1)*legendre(m, 3)),
+                mpf_sqrt(from_int(k//3), prec), prec),
+                mpf_sin(arg, prec), prec)
+        v = k + v % k
+        if v % p == 0:
+            if e == 1:
+                return mpf_mul(
+                    from_int(jacobi(3, k)),
+                    mpf_sqrt(from_int(k), prec), prec)
+            return fzero
+        if not is_quad_residue(v, p):
+            return fzero
+        _phi = p**(e - 1)*(p - 1)
+        v = (v*pow(576, _phi - 1, k))
+        m = _sqrt_mod_prime_power(v, p, e)[0]
+        arg = mpf_div(
+            mpf_mul(from_int(4*m), pi, prec),
+            from_int(k), prec)
+        return mpf_mul(mpf_mul(
+            from_int(2*jacobi(3, k)),
+            mpf_sqrt(from_int(k), prec), prec),
+            mpf_cos(arg, prec), prec)
+
+    if p != 2 or e >= 3:
+        d1, d2 = gcd(k1, 24), gcd(k2, 24)
+        e = 24//(d1*d2)
+        n1 = ((d2*e*n + (k2**2 - 1)//d1)*
+            pow(e*k2*k2*d2, _totient[k1] - 1, k1)) % k1
+        n2 = ((d1*e*n + (k1**2 - 1)//d2)*
+            pow(e*k1*k1*d1, _totient[k2] - 1, k2)) % k2
+        return mpf_mul(_a(n1, k1, prec), _a(n2, k2, prec), prec)
+    if e == 2:
+        n1 = ((8*n + 5)*pow(128, _totient[k1] - 1, k1)) % k1
+        n2 = (4 + ((n - 2 - (k1**2 - 1)//8)*(k1**2)) % 4) % 4
+        return mpf_mul(mpf_mul(
+            from_int(-1),
+            _a(n1, k1, prec), prec),
+            _a(n2, k2, prec))
+    n1 = ((8*n + 1)*pow(32, _totient[k1] - 1, k1)) % k1
+    n2 = (2 + (n - (k1**2 - 1)//8) % 2) % 2
+    return mpf_mul(_a(n1, k1, prec), _a(n2, k2, prec), prec)
+
+def _d(n, j, prec, sq23pi, sqrt8):
+    """
+    Compute the sinh term in the outer sum of the HRR formula.
+    The constants sqrt(2/3*pi) and sqrt(8) must be precomputed.
+    """
+    j = from_int(j)
+    pi = mpf_pi(prec)
+    a = mpf_div(sq23pi, j, prec)
+    b = mpf_sub(from_int(n), from_rational(1, 24, prec), prec)
+    c = mpf_sqrt(b, prec)
+    ch, sh = mpf_cosh_sinh(mpf_mul(a, c), prec)
+    D = mpf_div(
+        mpf_sqrt(j, prec),
+        mpf_mul(mpf_mul(sqrt8, b), pi), prec)
+    E = mpf_sub(mpf_mul(a, ch), mpf_div(sh, c, prec), prec)
+    return mpf_mul(D, E)
+
+
+@recurrence_memo([1, 1])
+def _partition_rec(n: int, prev) -> int:
+    """ Calculate the partition function P(n)
+
+    Parameters
+    ==========
+
+    n : int
+        nonnegative integer
+
+    """
+    v = 0
+    penta = 0 # pentagonal number: 1, 5, 12, ...
+    for i in count():
+        penta += 3*i + 1
+        np = n - penta
+        if np < 0:
+            break
+        s = prev[np]
+        np -= i + 1
+        # np = n - gp where gp = generalized pentagonal: 2, 7, 15, ...
+        if 0 <= np:
+            s += prev[np]
+        v += -s if i % 2 else s
+    return v
+
+
+def _partition(n: int) -> int:
+    """ Calculate the partition function P(n)
+
+    Parameters
+    ==========
+
+    n : int
+
+    """
+    if n < 0:
+        return 0
+    if (n <= 200_000 and n - _partition_rec.cache_length() < 70 or
+            _partition_rec.cache_length() == 2 and n < 14_400):
+        # There will be 2*10**5 elements created here
+        # and n elements created by partition, so in case we
+        # are going to be working with small n, we just
+        # use partition to calculate (and cache) the values
+        # since lookup is used there while summation, using
+        # _factor and _totient, will be used below. But we
+        # only do so if n is relatively close to the length
+        # of the cache since doing 1 calculation here is about
+        # the same as adding 70 elements to the cache. In addition,
+        # the startup here costs about the same as calculating the first
+        # 14,400 values via partition, so we delay startup here unless n
+        # is smaller than that.
+        return _partition_rec(n)
+    if '_factor' not in globals():
+        _pre()
+    # Estimate number of bits in p(n). This formula could be tidied
+    pbits = int((
+        math.pi*(2*n/3.)**0.5 -
+        math.log(4*n))/math.log(10) + 1) * \
+        math.log2(10)
+    prec = p = int(pbits*1.1 + 100)
+
+    # find the number of terms needed so rounded sum will be accurate
+    # using Rademacher's bound M(n, N) for the remainder after a partial
+    # sum of N terms (https://arxiv.org/pdf/1205.5991.pdf, (1.8))
+    c1 = 44*math.pi**2/(225*math.sqrt(3))
+    c2 = math.pi*math.sqrt(2)/75
+    c3 = math.pi*math.sqrt(2/3)
+    def _M(n, N):
+        sqrt = math.sqrt
+        return c1/sqrt(N) + c2*sqrt(N/(n - 1))*math.sinh(c3*sqrt(n)/N)
+    big = max(9, math.ceil(n**0.5))  # should be too large (for n > 65, ceil should work)
+    assert _M(n, big) < 0.5  # else double big until too large
+    while big > 40 and _M(n, big) < 0.5:
+        big //= 2
+    small = big
+    big = small*2
+    while big - small > 1:
+        N = (big + small)//2
+        if (er := _M(n, N)) < 0.5:
+            big = N
+        elif er >= 0.5:
+            small = N
+    M = big  # done with function M; now have value
+
+    # sanity check for expected size of answer
+    if M > 10**5:  # i.e. M > maxn
+        raise ValueError("Input too big")  # i.e. n > 149832547102
+
+    # calculate it
+    s = fzero
+    sq23pi = mpf_mul(mpf_sqrt(from_rational(2, 3, p), p), mpf_pi(p), p)
+    sqrt8 = mpf_sqrt(from_int(8), p)
+    for q in range(1, M):
+        a = _a(n, q, p)
+        d = _d(n, q, p, sq23pi, sqrt8)
+        s = mpf_add(s, mpf_mul(a, d), prec)
+        # On average, the terms decrease rapidly in magnitude.
+        # Dynamically reducing the precision greatly improves
+        # performance.
+        p = bitcount(abs(to_int(d))) + 50
+    return int(to_int(mpf_add(s, fhalf, prec)))
+
+
+@deprecated("""\
+The `sympy.ntheory.partitions_.npartitions` has been moved to `sympy.functions.combinatorial.numbers.partition`.""",
+deprecated_since_version="1.13",
+active_deprecations_target='deprecated-ntheory-symbolic-functions')
+def npartitions(n, verbose=False):
+    """
+    Calculate the partition function P(n), i.e. the number of ways that
+    n can be written as a sum of positive integers.
+
+    .. deprecated:: 1.13
+
+        The ``npartitions`` function is deprecated. Use :class:`sympy.functions.combinatorial.numbers.partition`
+        instead. See its documentation for more information. See
+        :ref:`deprecated-ntheory-symbolic-functions` for details.
+
+    P(n) is computed using the Hardy-Ramanujan-Rademacher formula [1]_.
+
+
+    The correctness of this implementation has been tested through $10^{10}$.
+
+    Examples
+    ========
+
+    >>> from sympy.functions.combinatorial.numbers import partition
+    >>> partition(25)
+    1958
+
+    References
+    ==========
+
+    .. [1] https://mathworld.wolfram.com/PartitionFunctionP.html
+
+    """
+    from sympy.functions.combinatorial.numbers import partition as func_partition
+    return func_partition(n)

rl/Lib/site-packages/sympy/ntheory/primetest.py

@@ -0,0 +1,793 @@
+"""
+Primality testing
+
+"""
+
+from itertools import count
+
+from sympy.core.sympify import sympify
+from sympy.external.gmpy import (gmpy as _gmpy, gcd, jacobi,
+                                 is_square as gmpy_is_square,
+                                 bit_scan1, is_fermat_prp, is_euler_prp,
+                                 is_selfridge_prp, is_strong_selfridge_prp,
+                                 is_strong_bpsw_prp)
+from sympy.external.ntheory import _lucas_sequence
+from sympy.utilities.misc import as_int, filldedent
+
+# Note: This list should be updated whenever new Mersenne primes are found.
+# Refer: https://www.mersenne.org/
+MERSENNE_PRIME_EXPONENTS = (2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279, 2203,
+ 2281, 3217, 4253, 4423, 9689, 9941, 11213, 19937, 21701, 23209, 44497, 86243, 110503, 132049,
+ 216091, 756839, 859433, 1257787, 1398269, 2976221, 3021377, 6972593, 13466917, 20996011, 24036583,
+ 25964951, 30402457, 32582657, 37156667, 42643801, 43112609, 57885161, 74207281, 77232917, 82589933)
+
+
+def is_fermat_pseudoprime(n, a):
+    r"""Returns True if ``n`` is prime or is an odd composite integer that
+    is coprime to ``a`` and satisfy the modular arithmetic congruence relation:
+
+    .. math ::
+        a^{n-1} \equiv 1 \pmod{n}
+
+    (where mod refers to the modulo operation).
+
+    Parameters
+    ==========
+
+    n : Integer
+        ``n`` is a positive integer.
+    a : Integer
+        ``a`` is a positive integer.
+        ``a`` and ``n`` should be relatively prime.
+
+    Returns
+    =======
+
+    bool : If ``n`` is prime, it always returns ``True``.
+           The composite number that returns ``True`` is called an Fermat pseudoprime.
+
+    Examples
+    ========
+
+    >>> from sympy.ntheory.primetest import is_fermat_pseudoprime
+    >>> from sympy.ntheory.factor_ import isprime
+    >>> for n in range(1, 1000):
+    ...     if is_fermat_pseudoprime(n, 2) and not isprime(n):
+    ...         print(n)
+    341
+    561
+    645
+
+    References
+    ==========
+
+    .. [1] https://en.wikipedia.org/wiki/Fermat_pseudoprime
+    """
+    n, a = as_int(n), as_int(a)
+    if a == 1:
+        return n == 2 or bool(n % 2)
+    return is_fermat_prp(n, a)
+
+
+def is_euler_pseudoprime(n, a):
+    r"""Returns True if ``n`` is prime or is an odd composite integer that
+    is coprime to ``a`` and satisfy the modular arithmetic congruence relation:
+
+    .. math ::
+        a^{(n-1)/2} \equiv \pm 1 \pmod{n}
+
+    (where mod refers to the modulo operation).
+
+    Parameters
+    ==========
+
+    n : Integer
+        ``n`` is a positive integer.
+    a : Integer
+        ``a`` is a positive integer.
+        ``a`` and ``n`` should be relatively prime.
+
+    Returns
+    =======
+
+    bool : If ``n`` is prime, it always returns ``True``.
+           The composite number that returns ``True`` is called an Euler pseudoprime.
+
+    Examples
+    ========
+
+    >>> from sympy.ntheory.primetest import is_euler_pseudoprime
+    >>> from sympy.ntheory.factor_ import isprime
+    >>> for n in range(1, 1000):
+    ...     if is_euler_pseudoprime(n, 2) and not isprime(n):
+    ...         print(n)
+    341
+    561
+
+    References
+    ==========
+
+    .. [1] https://en.wikipedia.org/wiki/Euler_pseudoprime
+    """
+    n, a = as_int(n), as_int(a)
+    if a < 1:
+        raise ValueError("a should be an integer greater than 0")
+    if n < 1:
+        raise ValueError("n should be an integer greater than 0")
+    if n == 1:
+        return False
+    if a == 1:
+        return n == 2 or bool(n % 2)  # (prime or odd composite)
+    if n % 2 == 0:
+        return n == 2
+    if gcd(n, a) != 1:
+        raise ValueError("The two numbers should be relatively prime")
+    return pow(a, (n - 1) // 2, n) in [1, n - 1]
+
+
+def is_euler_jacobi_pseudoprime(n, a):
+    r"""Returns True if ``n`` is prime or is an odd composite integer that
+    is coprime to ``a`` and satisfy the modular arithmetic congruence relation:
+
+    .. math ::
+        a^{(n-1)/2} \equiv \left(\frac{a}{n}\right) \pmod{n}
+
+    (where mod refers to the modulo operation).
+
+    Parameters
+    ==========
+
+    n : Integer
+        ``n`` is a positive integer.
+    a : Integer
+        ``a`` is a positive integer.
+        ``a`` and ``n`` should be relatively prime.
+
+    Returns
+    =======
+
+    bool : If ``n`` is prime, it always returns ``True``.
+           The composite number that returns ``True`` is called an Euler-Jacobi pseudoprime.
+
+    Examples
+    ========
+
+    >>> from sympy.ntheory.primetest import is_euler_jacobi_pseudoprime
+    >>> from sympy.ntheory.factor_ import isprime
+    >>> for n in range(1, 1000):
+    ...     if is_euler_jacobi_pseudoprime(n, 2) and not isprime(n):
+    ...         print(n)
+    561
+
+    References
+    ==========
+
+    .. [1] https://en.wikipedia.org/wiki/Euler%E2%80%93Jacobi_pseudoprime
+    """
+    n, a = as_int(n), as_int(a)
+    if a == 1:
+        return n == 2 or bool(n % 2)
+    return is_euler_prp(n, a)
+
+
+def is_square(n, prep=True):
+    """Return True if n == a * a for some integer a, else False.
+    If n is suspected of *not* being a square then this is a
+    quick method of confirming that it is not.
+
+    Examples
+    ========
+
+    >>> from sympy.ntheory.primetest import is_square
+    >>> is_square(25)
+    True
+    >>> is_square(2)
+    False
+
+    References
+    ==========
+
+    .. [1]  https://mersenneforum.org/showpost.php?p=110896
+
+    See Also
+    ========
+    sympy.core.intfunc.isqrt
+    """
+    if prep:
+        n = as_int(n)
+        if n < 0:
+            return False
+        if n in (0, 1):
+            return True
+    return gmpy_is_square(n)
+
+
+def _test(n, base, s, t):
+    """Miller-Rabin strong pseudoprime test for one base.
+    Return False if n is definitely composite, True if n is
+    probably prime, with a probability greater than 3/4.
+
+    """
+    # do the Fermat test
+    b = pow(base, t, n)
+    if b == 1 or b == n - 1:
+        return True
+    for _ in range(s - 1):
+        b = pow(b, 2, n)
+        if b == n - 1:
+            return True
+        # see I. Niven et al. "An Introduction to Theory of Numbers", page 78
+        if b == 1:
+            return False
+    return False
+
+
+def mr(n, bases):
+    """Perform a Miller-Rabin strong pseudoprime test on n using a
+    given list of bases/witnesses.
+
+    References
+    ==========
+
+    .. [1] Richard Crandall & Carl Pomerance (2005), "Prime Numbers:
+           A Computational Perspective", Springer, 2nd edition, 135-138
+
+    A list of thresholds and the bases they require are here:
+    https://en.wikipedia.org/wiki/Miller%E2%80%93Rabin_primality_test#Deterministic_variants
+
+    Examples
+    ========
+
+    >>> from sympy.ntheory.primetest import mr
+    >>> mr(1373651, [2, 3])
+    False
+    >>> mr(479001599, [31, 73])
+    True
+
+    """
+    from sympy.polys.domains import ZZ
+
+    n = as_int(n)
+    if n < 2:
+        return False
+    # remove powers of 2 from n-1 (= t * 2**s)
+    s = bit_scan1(n - 1)
+    t = n >> s
+    for base in bases:
+        # Bases >= n are wrapped, bases < 2 are invalid
+        if base >= n:
+            base %= n
+        if base >= 2:
+            base = ZZ(base)
+            if not _test(n, base, s, t):
+                return False
+    return True
+
+
+def _lucas_extrastrong_params(n):
+    """Calculates the "extra strong" parameters (D, P, Q) for n.
+
+    Parameters
+    ==========
+
+    n : int
+        positive odd integer
+
+    Returns
+    =======
+
+    D, P, Q: "extra strong" parameters.
+             ``(0, 0, 0)`` if we find a nontrivial divisor of ``n``.
+
+    Examples
+    ========
+
+    >>> from sympy.ntheory.primetest import _lucas_extrastrong_params
+    >>> _lucas_extrastrong_params(101)
+    (12, 4, 1)
+    >>> _lucas_extrastrong_params(15)
+    (0, 0, 0)
+
+    References
+    ==========
+    .. [1] OEIS A217719: Extra Strong Lucas Pseudoprimes
+           https://oeis.org/A217719
+    .. [2] https://en.wikipedia.org/wiki/Lucas_pseudoprime
+
+    """
+    for P in count(3):
+        D = P**2 - 4
+        j = jacobi(D, n)
+        if j == -1:
+            return (D, P, 1)
+        elif j == 0 and D % n:
+            return (0, 0, 0)
+
+
+def is_lucas_prp(n):
+    """Standard Lucas compositeness test with Selfridge parameters.  Returns
+    False if n is definitely composite, and True if n is a Lucas probable
+    prime.
+
+    This is typically used in combination with the Miller-Rabin test.
+
+    References
+    ==========
+    .. [1] Robert Baillie, Samuel S. Wagstaff, Lucas Pseudoprimes,
+           Math. Comp. Vol 35, Number 152 (1980), pp. 1391-1417,
+           https://doi.org/10.1090%2FS0025-5718-1980-0583518-6
+           http://mpqs.free.fr/LucasPseudoprimes.pdf
+    .. [2] OEIS A217120: Lucas Pseudoprimes
+           https://oeis.org/A217120
+    .. [3] https://en.wikipedia.org/wiki/Lucas_pseudoprime
+
+    Examples
+    ========
+
+    >>> from sympy.ntheory.primetest import isprime, is_lucas_prp
+    >>> for i in range(10000):
+    ...     if is_lucas_prp(i) and not isprime(i):
+    ...         print(i)
+    323
+    377
+    1159
+    1829
+    3827
+    5459
+    5777
+    9071
+    9179
+    """
+    n = as_int(n)
+    if n < 2:
+        return False
+    return is_selfridge_prp(n)
+
+
+def is_strong_lucas_prp(n):
+    """Strong Lucas compositeness test with Selfridge parameters.  Returns
+    False if n is definitely composite, and True if n is a strong Lucas
+    probable prime.
+
+    This is often used in combination with the Miller-Rabin test, and
+    in particular, when combined with M-R base 2 creates the strong BPSW test.
+
+    References
+    ==========
+    .. [1] Robert Baillie, Samuel S. Wagstaff, Lucas Pseudoprimes,
+           Math. Comp. Vol 35, Number 152 (1980), pp. 1391-1417,
+           https://doi.org/10.1090%2FS0025-5718-1980-0583518-6
+           http://mpqs.free.fr/LucasPseudoprimes.pdf
+    .. [2] OEIS A217255: Strong Lucas Pseudoprimes
+           https://oeis.org/A217255
+    .. [3] https://en.wikipedia.org/wiki/Lucas_pseudoprime
+    .. [4] https://en.wikipedia.org/wiki/Baillie-PSW_primality_test
+
+    Examples
+    ========
+
+    >>> from sympy.ntheory.primetest import isprime, is_strong_lucas_prp
+    >>> for i in range(20000):
+    ...     if is_strong_lucas_prp(i) and not isprime(i):
+    ...        print(i)
+    5459
+    5777
+    10877
+    16109
+    18971
+    """
+    n = as_int(n)
+    if n < 2:
+        return False
+    return is_strong_selfridge_prp(n)
+
+
+def is_extra_strong_lucas_prp(n):
+    """Extra Strong Lucas compositeness test.  Returns False if n is
+    definitely composite, and True if n is an "extra strong" Lucas probable
+    prime.
+
+    The parameters are selected using P = 3, Q = 1, then incrementing P until
+    (D|n) == -1.  The test itself is as defined in [1]_, from the
+    Mo and Jones preprint.  The parameter selection and test are the same as
+    used in OEIS A217719, Perl's Math::Prime::Util, and the Lucas pseudoprime
+    page on Wikipedia.
+
+    It is 20-50% faster than the strong test.
+
+    Because of the different parameters selected, there is no relationship
+    between the strong Lucas pseudoprimes and extra strong Lucas pseudoprimes.
+    In particular, one is not a subset of the other.
+
+    References
+    ==========
+    .. [1] Jon Grantham, Frobenius Pseudoprimes,
+           Math. Comp. Vol 70, Number 234 (2001), pp. 873-891,
+           https://doi.org/10.1090%2FS0025-5718-00-01197-2
+    .. [2] OEIS A217719: Extra Strong Lucas Pseudoprimes
+           https://oeis.org/A217719
+    .. [3] https://en.wikipedia.org/wiki/Lucas_pseudoprime
+
+    Examples
+    ========
+
+    >>> from sympy.ntheory.primetest import isprime, is_extra_strong_lucas_prp
+    >>> for i in range(20000):
+    ...     if is_extra_strong_lucas_prp(i) and not isprime(i):
+    ...        print(i)
+    989
+    3239
+    5777
+    10877
+    """
+    # Implementation notes:
+    #   1) the parameters differ from Thomas R. Nicely's.  His parameter
+    #      selection leads to pseudoprimes that overlap M-R tests, and
+    #      contradict Baillie and Wagstaff's suggestion of (D|n) = -1.
+    #   2) The MathWorld page as of June 2013 specifies Q=-1.  The Lucas
+    #      sequence must have Q=1.  See Grantham theorem 2.3, any of the
+    #      references on the MathWorld page, or run it and see Q=-1 is wrong.
+    n = as_int(n)
+    if n == 2:
+        return True
+    if n < 2 or (n % 2) == 0:
+        return False
+    if gmpy_is_square(n):
+        return False
+
+    D, P, Q = _lucas_extrastrong_params(n)
+    if D == 0:
+        return False
+
+    # remove powers of 2 from n+1 (= k * 2**s)
+    s = bit_scan1(n + 1)
+    k = (n + 1) >> s
+
+    U, V, _ = _lucas_sequence(n, P, Q, k)
+
+    if U == 0 and (V == 2 or V == n - 2):
+        return True
+    for _ in range(1, s):
+        if V == 0:
+            return True
+        V = (V*V - 2) % n
+    return False
+
+
+def proth_test(n):
+    r""" Test if the Proth number `n = k2^m + 1` is prime. where k is a positive odd number and `2^m > k`.
+
+    Parameters
+    ==========
+
+    n : Integer
+        ``n`` is Proth number
+
+    Returns
+    =======
+
+    bool : If ``True``, then ``n`` is the Proth prime
+
+    Raises
+    ======
+
+    ValueError
+        If ``n`` is not Proth number.
+
+    Examples
+    ========
+
+    >>> from sympy.ntheory.primetest import proth_test
+    >>> proth_test(41)
+    True
+    >>> proth_test(57)
+    False
+
+    References
+    ==========
+
+    .. [1] https://en.wikipedia.org/wiki/Proth_prime
+
+    """
+    n = as_int(n)
+    if n < 3:
+        raise ValueError("n is not Proth number")
+    m = bit_scan1(n - 1)
+    k = n >> m
+    if m < k.bit_length():
+        raise ValueError("n is not Proth number")
+    if n % 3 == 0:
+        return n == 3
+    if k % 3: # n % 12 == 5
+        return pow(3, n >> 1, n) == n - 1
+    # If `n` is a square number, then `jacobi(a, n) = 1` for any `a`
+    if gmpy_is_square(n):
+        return False
+    # `a` may be chosen at random.
+    # In any case, we want to find `a` such that `jacobi(a, n) = -1`.
+    for a in range(5, n):
+        j = jacobi(a, n)
+        if j == -1:
+            return pow(a, n >> 1, n) == n - 1
+        if j == 0:
+            return False
+
+
+def _lucas_lehmer_primality_test(p):
+    r""" Test if the Mersenne number `M_p = 2^p-1` is prime.
+
+    Parameters
+    ==========
+
+    p : int
+        ``p`` is an odd prime number
+
+    Returns
+    =======
+
+    bool : If ``True``, then `M_p` is the Mersenne prime
+
+    Examples
+    ========
+
+    >>> from sympy.ntheory.primetest import _lucas_lehmer_primality_test
+    >>> _lucas_lehmer_primality_test(5) # 2**5 - 1 = 31 is prime
+    True
+    >>> _lucas_lehmer_primality_test(11) # 2**11 - 1 = 2047 is not prime
+    False
+
+    See Also
+    ========
+
+    is_mersenne_prime
+
+    References
+    ==========
+
+    .. [1] https://en.wikipedia.org/wiki/Lucas%E2%80%93Lehmer_primality_test
+
+    """
+    v = 4
+    m = 2**p - 1
+    for _ in range(p - 2):
+        v = pow(v, 2, m) - 2
+    return v == 0
+
+
+def is_mersenne_prime(n):
+    """Returns True if  ``n`` is a Mersenne prime, else False.
+
+    A Mersenne prime is a prime number having the form `2^i - 1`.
+
+    Examples
+    ========
+
+    >>> from sympy.ntheory.factor_ import is_mersenne_prime
+    >>> is_mersenne_prime(6)
+    False
+    >>> is_mersenne_prime(127)
+    True
+
+    References
+    ==========
+
+    .. [1] https://mathworld.wolfram.com/MersennePrime.html
+
+    """
+    n = as_int(n)
+    if n < 1:
+        return False
+    if n & (n + 1):
+        # n is not Mersenne number
+        return False
+    p = n.bit_length()
+    if p in MERSENNE_PRIME_EXPONENTS:
+        return True
+    if p < 65_000_000 or not isprime(p):
+        # According to GIMPS, verification was completed on September 19, 2023 for p less than 65 million.
+        # https://www.mersenne.org/report_milestones/
+        # If p is composite number, then n=2**p-1 is composite number.
+        return False
+    result = _lucas_lehmer_primality_test(p)
+    if result:
+        raise ValueError(filldedent('''
+            This Mersenne Prime, 2^%s - 1, should
+            be added to SymPy's known values.''' % p))
+    return result
+
+
+def isprime(n):
+    """
+    Test if n is a prime number (True) or not (False). For n < 2^64 the
+    answer is definitive; larger n values have a small probability of actually
+    being pseudoprimes.
+
+    Negative numbers (e.g. -2) are not considered prime.
+
+    The first step is looking for trivial factors, which if found enables
+    a quick return.  Next, if the sieve is large enough, use bisection search
+    on the sieve.  For small numbers, a set of deterministic Miller-Rabin
+    tests are performed with bases that are known to have no counterexamples
+    in their range.  Finally if the number is larger than 2^64, a strong
+    BPSW test is performed.  While this is a probable prime test and we
+    believe counterexamples exist, there are no known counterexamples.
+
+    Examples
+    ========
+
+    >>> from sympy.ntheory import isprime
+    >>> isprime(13)
+    True
+    >>> isprime(15)
+    False
+
+    Notes
+    =====
+
+    This routine is intended only for integer input, not numerical
+    expressions which may represent numbers. Floats are also
+    rejected as input because they represent numbers of limited
+    precision. While it is tempting to permit 7.0 to represent an
+    integer there are errors that may "pass silently" if this is
+    allowed:
+
+    >>> from sympy import Float, S
+    >>> int(1e3) == 1e3 == 10**3
+    True
+    >>> int(1e23) == 1e23
+    True
+    >>> int(1e23) == 10**23
+    False
+
+    >>> near_int = 1 + S(1)/10**19
+    >>> near_int == int(near_int)
+    False
+    >>> n = Float(near_int, 10)  # truncated by precision
+    >>> n % 1 == 0
+    True
+    >>> n = Float(near_int, 20)
+    >>> n % 1 == 0
+    False
+
+    See Also
+    ========
+
+    sympy.ntheory.generate.primerange : Generates all primes in a given range
+    sympy.functions.combinatorial.numbers.primepi : Return the number of primes less than or equal to n
+    sympy.ntheory.generate.prime : Return the nth prime
+
+    References
+    ==========
+    .. [1] https://en.wikipedia.org/wiki/Strong_pseudoprime
+    .. [2] Robert Baillie, Samuel S. Wagstaff, Lucas Pseudoprimes,
+           Math. Comp. Vol 35, Number 152 (1980), pp. 1391-1417,
+           https://doi.org/10.1090%2FS0025-5718-1980-0583518-6
+           http://mpqs.free.fr/LucasPseudoprimes.pdf
+    .. [3] https://en.wikipedia.org/wiki/Baillie-PSW_primality_test
+    """
+    n = as_int(n)
+
+    # Step 1, do quick composite testing via trial division.  The individual
+    # modulo tests benchmark faster than one or two primorial igcds for me.
+    # The point here is just to speedily handle small numbers and many
+    # composites.  Step 2 only requires that n <= 2 get handled here.
+    if n in [2, 3, 5]:
+        return True
+    if n < 2 or (n % 2) == 0 or (n % 3) == 0 or (n % 5) == 0:
+        return False
+    if n < 49:
+        return True
+    if (n %  7) == 0 or (n % 11) == 0 or (n % 13) == 0 or (n % 17) == 0 or \
+       (n % 19) == 0 or (n % 23) == 0 or (n % 29) == 0 or (n % 31) == 0 or \
+       (n % 37) == 0 or (n % 41) == 0 or (n % 43) == 0 or (n % 47) == 0:
+        return False
+    if n < 2809:
+        return True
+    if n < 65077:
+        # There are only five Euler pseudoprimes with a least prime factor greater than 47
+        return pow(2, n >> 1, n) in [1, n - 1] and n not in [8321, 31621, 42799, 49141, 49981]
+
+    # bisection search on the sieve if the sieve is large enough
+    from sympy.ntheory.generate import sieve as s
+    if n <= s._list[-1]:
+        l, u = s.search(n)
+        return l == u
+
+    # If we have GMPY2, skip straight to step 3 and do a strong BPSW test.
+    # This should be a bit faster than our step 2, and for large values will
+    # be a lot faster than our step 3 (C+GMP vs. Python).
+    if _gmpy is not None:
+        return is_strong_bpsw_prp(n)
+
+
+    # Step 2: deterministic Miller-Rabin testing for numbers < 2^64.  See:
+    #    https://miller-rabin.appspot.com/
+    # for lists.  We have made sure the M-R routine will successfully handle
+    # bases larger than n, so we can use the minimal set.
+    # In September 2015 deterministic numbers were extended to over 2^81.
+    #    https://arxiv.org/pdf/1509.00864.pdf
+    #    https://oeis.org/A014233
+    if n < 341531:
+        return mr(n, [9345883071009581737])
+    if n < 885594169:
+        return mr(n, [725270293939359937, 3569819667048198375])
+    if n < 350269456337:
+        return mr(n, [4230279247111683200, 14694767155120705706, 16641139526367750375])
+    if n < 55245642489451:
+        return mr(n, [2, 141889084524735, 1199124725622454117, 11096072698276303650])
+    if n < 7999252175582851:
+        return mr(n, [2, 4130806001517, 149795463772692060, 186635894390467037, 3967304179347715805])
+    if n < 585226005592931977:
+        return mr(n, [2, 123635709730000, 9233062284813009, 43835965440333360, 761179012939631437, 1263739024124850375])
+    if n < 18446744073709551616:
+        return mr(n, [2, 325, 9375, 28178, 450775, 9780504, 1795265022])
+    if n < 318665857834031151167461:
+        return mr(n, [2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37])
+    if n < 3317044064679887385961981:
+        return mr(n, [2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41])
+
+    # We could do this instead at any point:
+    #if n < 18446744073709551616:
+    #   return mr(n, [2]) and is_extra_strong_lucas_prp(n)
+
+    # Here are tests that are safe for MR routines that don't understand
+    # large bases.
+    #if n < 9080191:
+    #    return mr(n, [31, 73])
+    #if n < 19471033:
+    #    return mr(n, [2, 299417])
+    #if n < 38010307:
+    #    return mr(n, [2, 9332593])
+    #if n < 316349281:
+    #    return mr(n, [11000544, 31481107])
+    #if n < 4759123141:
+    #    return mr(n, [2, 7, 61])
+    #if n < 105936894253:
+    #    return mr(n, [2, 1005905886, 1340600841])
+    #if n < 31858317218647:
+    #    return mr(n, [2, 642735, 553174392, 3046413974])
+    #if n < 3071837692357849:
+    #    return mr(n, [2, 75088, 642735, 203659041, 3613982119])
+    #if n < 18446744073709551616:
+    #    return mr(n, [2, 325, 9375, 28178, 450775, 9780504, 1795265022])
+
+    # Step 3: BPSW.
+    #
+    #  Time for isprime(10**2000 + 4561), no gmpy or gmpy2 installed
+    #     44.0s   old isprime using 46 bases
+    #      5.3s   strong BPSW + one random base
+    #      4.3s   extra strong BPSW + one random base
+    #      4.1s   strong BPSW
+    #      3.2s   extra strong BPSW
+
+    # Classic BPSW from page 1401 of the paper.  See alternate ideas below.
+    return is_strong_bpsw_prp(n)
+
+    # Using extra strong test, which is somewhat faster
+    #return mr(n, [2]) and is_extra_strong_lucas_prp(n)
+
+    # Add a random M-R base
+    #import random
+    #return mr(n, [2, random.randint(3, n-1)]) and is_strong_lucas_prp(n)
+
+
+def is_gaussian_prime(num):
+    r"""Test if num is a Gaussian prime number.
+
+    References
+    ==========
+
+    .. [1] https://oeis.org/wiki/Gaussian_primes
+    """
+
+    num = sympify(num)
+    a, b = num.as_real_imag()
+    a = as_int(a, strict=False)
+    b = as_int(b, strict=False)
+    if a == 0:
+        b = abs(b)
+        return isprime(b) and b % 4 == 3
+    elif b == 0:
+        a = abs(a)
+        return isprime(a) and a % 4 == 3
+    return isprime(a**2 + b**2)

rl/Lib/site-packages/sympy/ntheory/qs.py

@@ -0,0 +1,511 @@
+from sympy.core.random import _randint
+from sympy.external.gmpy import gcd, invert, sqrt as isqrt
+from sympy.ntheory.residue_ntheory import _sqrt_mod_prime_power
+from sympy.ntheory import isprime
+from math import log, sqrt
+
+
+class SievePolynomial:
+    def __init__(self, modified_coeff=(), a=None, b=None):
+        """This class denotes the seive polynomial.
+        If ``g(x) = (a*x + b)**2 - N``. `g(x)` can be expanded
+        to ``a*x**2 + 2*a*b*x + b**2 - N``, so the coefficient
+        is stored in the form `[a**2, 2*a*b, b**2 - N]`. This
+        ensures faster `eval` method because we dont have to
+        perform `a**2, 2*a*b, b**2` every time we call the
+        `eval` method. As multiplication is more expensive
+        than addition, by using modified_coefficient we get
+        a faster seiving process.
+
+        Parameters
+        ==========
+
+        modified_coeff : modified_coefficient of sieve polynomial
+        a : parameter of the sieve polynomial
+        b : parameter of the sieve polynomial
+        """
+        self.modified_coeff = modified_coeff
+        self.a = a
+        self.b = b
+
+    def eval(self, x):
+        """
+        Compute the value of the sieve polynomial at point x.
+
+        Parameters
+        ==========
+
+        x : Integer parameter for sieve polynomial
+        """
+        ans = 0
+        for coeff in self.modified_coeff:
+            ans *= x
+            ans += coeff
+        return ans
+
+
+class FactorBaseElem:
+    """This class stores an element of the `factor_base`.
+    """
+    def __init__(self, prime, tmem_p, log_p):
+        """
+        Initialization of factor_base_elem.
+
+        Parameters
+        ==========
+
+        prime : prime number of the factor_base
+        tmem_p : Integer square root of x**2 = n mod prime
+        log_p : Compute Natural Logarithm of the prime
+        """
+        self.prime = prime
+        self.tmem_p = tmem_p
+        self.log_p = log_p
+        self.soln1 = None
+        self.soln2 = None
+        self.a_inv = None
+        self.b_ainv = None
+
+
+def _generate_factor_base(prime_bound, n):
+    """Generate `factor_base` for Quadratic Sieve. The `factor_base`
+    consists of all the points whose ``legendre_symbol(n, p) == 1``
+    and ``p < num_primes``. Along with the prime `factor_base` also stores
+    natural logarithm of prime and the residue n modulo p.
+    It also returns the of primes numbers in the `factor_base` which are
+    close to 1000 and 5000.
+
+    Parameters
+    ==========
+
+    prime_bound : upper prime bound of the factor_base
+    n : integer to be factored
+    """
+    from sympy.ntheory.generate import sieve
+    factor_base = []
+    idx_1000, idx_5000 = None, None
+    for prime in sieve.primerange(1, prime_bound):
+        if pow(n, (prime - 1) // 2, prime) == 1:
+            if prime > 1000 and idx_1000 is None:
+                idx_1000 = len(factor_base) - 1
+            if prime > 5000 and idx_5000 is None:
+                idx_5000 = len(factor_base) - 1
+            residue = _sqrt_mod_prime_power(n, prime, 1)[0]
+            log_p = round(log(prime)*2**10)
+            factor_base.append(FactorBaseElem(prime, residue, log_p))
+    return idx_1000, idx_5000, factor_base
+
+
+def _initialize_first_polynomial(N, M, factor_base, idx_1000, idx_5000, seed=None):
+    """This step is the initialization of the 1st sieve polynomial.
+    Here `a` is selected as a product of several primes of the factor_base
+    such that `a` is about to ``sqrt(2*N) / M``. Other initial values of
+    factor_base elem are also initialized which includes a_inv, b_ainv, soln1,
+    soln2 which are used when the sieve polynomial is changed. The b_ainv
+    is required for fast polynomial change as we do not have to calculate
+    `2*b*invert(a, prime)` every time.
+    We also ensure that the `factor_base` primes which make `a` are between
+    1000 and 5000.
+
+    Parameters
+    ==========
+
+    N : Number to be factored
+    M : sieve interval
+    factor_base : factor_base primes
+    idx_1000 : index of prime number in the factor_base near 1000
+    idx_5000 : index of prime number in the factor_base near to 5000
+    seed : Generate pseudoprime numbers
+    """
+    randint = _randint(seed)
+    approx_val = sqrt(2*N) / M
+    # `a` is a parameter of the sieve polynomial and `q` is the prime factors of `a`
+    # randomly search for a combination of primes whose multiplication is close to approx_val
+    # This multiplication of primes will be `a` and the primes will be `q`
+    # `best_a` denotes that `a` is close to approx_val in the random search of combination
+    best_a, best_q, best_ratio = None, None, None
+    start = 0 if idx_1000 is None else idx_1000
+    end = len(factor_base) - 1 if idx_5000 is None else idx_5000
+    for _ in range(50):
+        a = 1
+        q = []
+        while(a < approx_val):
+            rand_p = 0
+            while(rand_p == 0 or rand_p in q):
+                rand_p = randint(start, end)
+            p = factor_base[rand_p].prime
+            a *= p
+            q.append(rand_p)
+        ratio = a / approx_val
+        if best_ratio is None or abs(ratio - 1) < abs(best_ratio - 1):
+            best_q = q
+            best_a = a
+            best_ratio = ratio
+
+    a = best_a
+    q = best_q
+
+    B = []
+    for val in q:
+        q_l = factor_base[val].prime
+        gamma = factor_base[val].tmem_p * invert(a // q_l, q_l) % q_l
+        if gamma > q_l / 2:
+            gamma = q_l - gamma
+        B.append(a//q_l*gamma)
+
+    b = sum(B)
+    g = SievePolynomial([a*a, 2*a*b, b*b - N], a, b)
+
+    for fb in factor_base:
+        if a % fb.prime == 0:
+            continue
+        fb.a_inv = invert(a, fb.prime)
+        fb.b_ainv = [2*b_elem*fb.a_inv % fb.prime for b_elem in B]
+        fb.soln1 = (fb.a_inv*(fb.tmem_p - b)) % fb.prime
+        fb.soln2 = (fb.a_inv*(-fb.tmem_p - b)) % fb.prime
+    return g, B
+
+
+def _initialize_ith_poly(N, factor_base, i, g, B):
+    """Initialization stage of ith poly. After we finish sieving 1`st polynomial
+    here we quickly change to the next polynomial from which we will again
+    start sieving. Suppose we generated ith sieve polynomial and now we
+    want to generate (i + 1)th polynomial, where ``1 <= i <= 2**(j - 1) - 1``
+    where `j` is the number of prime factors of the coefficient `a`
+    then this function can be used to go to the next polynomial. If
+    ``i = 2**(j - 1) - 1`` then go to _initialize_first_polynomial stage.
+
+    Parameters
+    ==========
+
+    N : number to be factored
+    factor_base : factor_base primes
+    i : integer denoting ith polynomial
+    g : (i - 1)th polynomial
+    B : array that stores a//q_l*gamma
+    """
+    from sympy.functions.elementary.integers import ceiling
+    v = 1
+    j = i
+    while(j % 2 == 0):
+        v += 1
+        j //= 2
+    if ceiling(i / (2**v)) % 2 == 1:
+        neg_pow = -1
+    else:
+        neg_pow = 1
+    b = g.b + 2*neg_pow*B[v - 1]
+    a = g.a
+    g = SievePolynomial([a*a, 2*a*b, b*b - N], a, b)
+    for fb in factor_base:
+        if a % fb.prime == 0:
+            continue
+        fb.soln1 = (fb.soln1 - neg_pow*fb.b_ainv[v - 1]) % fb.prime
+        fb.soln2 = (fb.soln2 - neg_pow*fb.b_ainv[v - 1]) % fb.prime
+
+    return g
+
+
+def _gen_sieve_array(M, factor_base):
+    """Sieve Stage of the Quadratic Sieve. For every prime in the factor_base
+    that does not divide the coefficient `a` we add log_p over the sieve_array
+    such that ``-M <= soln1 + i*p <=  M`` and ``-M <= soln2 + i*p <=  M`` where `i`
+    is an integer. When p = 2 then log_p is only added using
+    ``-M <= soln1 + i*p <=  M``.
+
+    Parameters
+    ==========
+
+    M : sieve interval
+    factor_base : factor_base primes
+    """
+    sieve_array = [0]*(2*M + 1)
+    for factor in factor_base:
+        if factor.soln1 is None: #The prime does not divides a
+            continue
+        for idx in range((M + factor.soln1) % factor.prime, 2*M, factor.prime):
+            sieve_array[idx] += factor.log_p
+        if factor.prime == 2:
+            continue
+        #if prime is 2 then sieve only with soln_1_p
+        for idx in range((M + factor.soln2) % factor.prime, 2*M, factor.prime):
+            sieve_array[idx] += factor.log_p
+    return sieve_array
+
+
+def _check_smoothness(num, factor_base):
+    """Here we check that if `num` is a smooth number or not. If `a` is a smooth
+    number then it returns a vector of prime exponents modulo 2. For example
+    if a = 2 * 5**2 * 7**3 and the factor base contains {2, 3, 5, 7} then
+    `a` is a smooth number and this function returns ([1, 0, 0, 1], True). If
+    `a` is a partial relation which means that `a` a has one prime factor
+    greater than the `factor_base` then it returns `(a, False)` which denotes `a`
+    is a partial relation.
+
+    Parameters
+    ==========
+
+    a : integer whose smootheness is to be checked
+    factor_base : factor_base primes
+    """
+    vec = []
+    if num < 0:
+        vec.append(1)
+        num *= -1
+    else:
+        vec.append(0)
+     #-1 is not included in factor_base add -1 in vector
+    for factor in factor_base:
+        if num % factor.prime != 0:
+            vec.append(0)
+            continue
+        factor_exp = 0
+        while num % factor.prime == 0:
+            factor_exp += 1
+            num //= factor.prime
+        vec.append(factor_exp % 2)
+    if num == 1:
+        return vec, True
+    if isprime(num):
+        return num, False
+    return None, None
+
+
+def _trial_division_stage(N, M, factor_base, sieve_array, sieve_poly, partial_relations, ERROR_TERM):
+    """Trial division stage. Here we trial divide the values generetated
+    by sieve_poly in the sieve interval and if it is a smooth number then
+    it is stored in `smooth_relations`. Moreover, if we find two partial relations
+    with same large prime then they are combined to form a smooth relation.
+    First we iterate over sieve array and look for values which are greater
+    than accumulated_val, as these values have a high chance of being smooth
+    number. Then using these values we find smooth relations.
+    In general, let ``t**2 = u*p modN`` and ``r**2 = v*p modN`` be two partial relations
+    with the same large prime p. Then they can be combined ``(t*r/p)**2 = u*v modN``
+    to form a smooth relation.
+
+    Parameters
+    ==========
+
+    N : Number to be factored
+    M : sieve interval
+    factor_base : factor_base primes
+    sieve_array : stores log_p values
+    sieve_poly : polynomial from which we find smooth relations
+    partial_relations : stores partial relations with one large prime
+    ERROR_TERM : error term for accumulated_val
+    """
+    sqrt_n = isqrt(N)
+    accumulated_val = log(M * sqrt_n)*2**10 - ERROR_TERM
+    smooth_relations = []
+    proper_factor = set()
+    partial_relation_upper_bound = 128*factor_base[-1].prime
+    for idx, val in enumerate(sieve_array):
+        if val < accumulated_val:
+            continue
+        x = idx - M
+        v = sieve_poly.eval(x)
+        vec, is_smooth = _check_smoothness(v, factor_base)
+        if is_smooth is None:#Neither smooth nor partial
+            continue
+        u = sieve_poly.a*x + sieve_poly.b
+        # Update the partial relation
+        # If 2 partial relation with same large prime is found then generate smooth relation
+        if is_smooth is False:#partial relation found
+            large_prime = vec
+            #Consider the large_primes under 128*F
+            if large_prime > partial_relation_upper_bound:
+                continue
+            if large_prime not in partial_relations:
+                partial_relations[large_prime] = (u, v)
+                continue
+            else:
+                u_prev, v_prev = partial_relations[large_prime]
+                partial_relations.pop(large_prime)
+                try:
+                    large_prime_inv = invert(large_prime, N)
+                except ZeroDivisionError:#if large_prime divides N
+                    proper_factor.add(large_prime)
+                    continue
+                u = u*u_prev*large_prime_inv
+                v = v*v_prev // (large_prime*large_prime)
+                vec, is_smooth = _check_smoothness(v, factor_base)
+        #assert u*u % N == v % N
+        smooth_relations.append((u, v, vec))
+    return smooth_relations, proper_factor
+
+
+#LINEAR ALGEBRA STAGE
+def _build_matrix(smooth_relations):
+    """Build a 2D matrix from smooth relations.
+
+    Parameters
+    ==========
+
+    smooth_relations : Stores smooth relations
+    """
+    matrix = []
+    for s_relation in smooth_relations:
+        matrix.append(s_relation[2])
+    return matrix
+
+
+def _gauss_mod_2(A):
+    """Fast gaussian reduction for modulo 2 matrix.
+
+    Parameters
+    ==========
+
+    A : Matrix
+
+    Examples
+    ========
+
+    >>> from sympy.ntheory.qs import _gauss_mod_2
+    >>> _gauss_mod_2([[0, 1, 1], [1, 0, 1], [0, 1, 0], [1, 1, 1]])
+    ([[[1, 0, 1], 3]],
+     [True, True, True, False],
+     [[0, 1, 0], [1, 0, 0], [0, 0, 1], [1, 0, 1]])
+
+    Reference
+    ==========
+
+    .. [1] A fast algorithm for gaussian elimination over GF(2) and
+    its implementation on the GAPP. Cetin K.Koc, Sarath N.Arachchige"""
+    import copy
+    matrix = copy.deepcopy(A)
+    row = len(matrix)
+    col = len(matrix[0])
+    mark = [False]*row
+    for c in range(col):
+        for r in range(row):
+            if matrix[r][c] == 1:
+                break
+        mark[r] = True
+        for c1 in range(col):
+            if c1 == c:
+                continue
+            if matrix[r][c1] == 1:
+                for r2 in range(row):
+                    matrix[r2][c1] = (matrix[r2][c1] + matrix[r2][c]) % 2
+    dependent_row = []
+    for idx, val in enumerate(mark):
+        if val == False:
+            dependent_row.append([matrix[idx], idx])
+    return dependent_row, mark, matrix
+
+
+def _find_factor(dependent_rows, mark, gauss_matrix, index, smooth_relations, N):
+    """Finds proper factor of N. Here, transform the dependent rows as a
+    combination of independent rows of the gauss_matrix to form the desired
+    relation of the form ``X**2 = Y**2 modN``. After obtaining the desired relation
+    we obtain a proper factor of N by `gcd(X - Y, N)`.
+
+    Parameters
+    ==========
+
+    dependent_rows : denoted dependent rows in the reduced matrix form
+    mark : boolean array to denoted dependent and independent rows
+    gauss_matrix : Reduced form of the smooth relations matrix
+    index : denoted the index of the dependent_rows
+    smooth_relations : Smooth relations vectors matrix
+    N : Number to be factored
+    """
+    idx_in_smooth = dependent_rows[index][1]
+    independent_u = [smooth_relations[idx_in_smooth][0]]
+    independent_v = [smooth_relations[idx_in_smooth][1]]
+    dept_row = dependent_rows[index][0]
+
+    for idx, val in enumerate(dept_row):
+        if val == 1:
+            for row in range(len(gauss_matrix)):
+                if gauss_matrix[row][idx] == 1 and mark[row] == True:
+                    independent_u.append(smooth_relations[row][0])
+                    independent_v.append(smooth_relations[row][1])
+                    break
+
+    u = 1
+    v = 1
+    for i in independent_u:
+        u *= i
+    for i in independent_v:
+        v *= i
+    #assert u**2 % N == v % N
+    v = isqrt(v)
+    return gcd(u - v, N)
+
+
+def qs(N, prime_bound, M, ERROR_TERM=25, seed=1234):
+    """Performs factorization using Self-Initializing Quadratic Sieve.
+    In SIQS, let N be a number to be factored, and this N should not be a
+    perfect power. If we find two integers such that ``X**2 = Y**2 modN`` and
+    ``X != +-Y modN``, then `gcd(X + Y, N)` will reveal a proper factor of N.
+    In order to find these integers X and Y we try to find relations of form
+    t**2 = u modN where u is a product of small primes. If we have enough of
+    these relations then we can form ``(t1*t2...ti)**2 = u1*u2...ui modN`` such that
+    the right hand side is a square, thus we found a relation of ``X**2 = Y**2 modN``.
+
+    Here, several optimizations are done like using multiple polynomials for
+    sieving, fast changing between polynomials and using partial relations.
+    The use of partial relations can speeds up the factoring by 2 times.
+
+    Parameters
+    ==========
+
+    N : Number to be Factored
+    prime_bound : upper bound for primes in the factor base
+    M : Sieve Interval
+    ERROR_TERM : Error term for checking smoothness
+    threshold : Extra smooth relations for factorization
+    seed : generate pseudo prime numbers
+
+    Examples
+    ========
+
+    >>> from sympy.ntheory import qs
+    >>> qs(25645121643901801, 2000, 10000)
+    {5394769, 4753701529}
+    >>> qs(9804659461513846513, 2000, 10000)
+    {4641991, 2112166839943}
+
+    References
+    ==========
+
+    .. [1] https://pdfs.semanticscholar.org/5c52/8a975c1405bd35c65993abf5a4edb667c1db.pdf
+    .. [2] https://www.rieselprime.de/ziki/Self-initializing_quadratic_sieve
+    """
+    ERROR_TERM*=2**10
+    idx_1000, idx_5000, factor_base = _generate_factor_base(prime_bound, N)
+    smooth_relations = []
+    ith_poly = 0
+    partial_relations = {}
+    proper_factor = set()
+    threshold = 5*len(factor_base) // 100
+    while True:
+        if ith_poly == 0:
+            ith_sieve_poly, B_array = _initialize_first_polynomial(N, M, factor_base, idx_1000, idx_5000)
+        else:
+            ith_sieve_poly = _initialize_ith_poly(N, factor_base, ith_poly, ith_sieve_poly, B_array)
+        ith_poly += 1
+        if ith_poly >= 2**(len(B_array) - 1): # time to start with a new sieve polynomial
+            ith_poly = 0
+        sieve_array = _gen_sieve_array(M, factor_base)
+        s_rel, p_f = _trial_division_stage(N, M, factor_base, sieve_array, ith_sieve_poly, partial_relations, ERROR_TERM)
+        smooth_relations += s_rel
+        proper_factor |= p_f
+        if len(smooth_relations) >= len(factor_base) + threshold:
+            break
+    matrix = _build_matrix(smooth_relations)
+    dependent_row, mark, gauss_matrix = _gauss_mod_2(matrix)
+    N_copy = N
+    for index in range(len(dependent_row)):
+        factor = _find_factor(dependent_row, mark, gauss_matrix, index, smooth_relations, N)
+        if factor > 1 and factor < N:
+            proper_factor.add(factor)
+            while(N_copy % factor == 0):
+                N_copy //= factor
+            if isprime(N_copy):
+                proper_factor.add(N_copy)
+                break
+            if(N_copy == 1):
+                break
+    return proper_factor

rl/Lib/site-packages/sympy/ntheory/tests/test_bbp_pi.py

@@ -0,0 +1,134 @@
+from sympy.core.random import randint
+
+from sympy.ntheory.bbp_pi import pi_hex_digits
+from sympy.testing.pytest import raises
+
+
+# http://www.herongyang.com/Cryptography/Blowfish-First-8366-Hex-Digits-of-PI.html
+# There are actually 8336 listed there; with the prepended 3 there are 8337
+# below
+dig=''.join('''
+3243f6a8885a308d313198a2e03707344a4093822299f31d0082efa98ec4e6c89452821e638d013
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+16636920d871574e69a458fea3f4933d7e0d95748f728eb658718bcd5882154aee7b54a41dc25a59
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+5b578fdfe33ac372e6'''.split())
+
+
+def test_hex_pi_nth_digits():
+    assert pi_hex_digits(0) == '3243f6a8885a30'
+    assert pi_hex_digits(1) ==  '243f6a8885a308'
+    assert pi_hex_digits(10000) == '68ac8fcfb8016c'
+    assert pi_hex_digits(13) == '08d313198a2e03'
+    assert pi_hex_digits(0, 3) == '324'
+    assert pi_hex_digits(0, 0) == ''
+    raises(ValueError, lambda: pi_hex_digits(-1))
+    raises(ValueError, lambda: pi_hex_digits(0, -1))
+    raises(ValueError, lambda: pi_hex_digits(3.14))
+
+    # this will pick a random segment to compute every time
+    # it is run. If it ever fails, there is an error in the
+    # computation.
+    n = randint(0, len(dig))
+    prec = randint(0, len(dig) - n)
+    assert pi_hex_digits(n, prec) == dig[n: n + prec]

rl/Lib/site-packages/sympy/ntheory/tests/test_continued_fraction.py

@@ -0,0 +1,77 @@
+import itertools
+from sympy.core import GoldenRatio as phi
+from sympy.core.numbers import (Rational, pi)
+from sympy.core.singleton import S
+from sympy.functions.elementary.miscellaneous import sqrt
+from sympy.ntheory.continued_fraction import \
+    (continued_fraction_periodic as cf_p,
+     continued_fraction_iterator as cf_i,
+     continued_fraction_convergents as cf_c,
+     continued_fraction_reduce as cf_r,
+     continued_fraction as cf)
+from sympy.testing.pytest import raises
+
+
+def test_continued_fraction():
+    assert cf_p(1, 1, 10, 0) == cf_p(1, 1, 0, 1)
+    assert cf_p(1, -1, 10, 1) == cf_p(-1, 1, 10, -1)
+    t = sqrt(2)
+    assert cf((1 + t)*(1 - t)) == cf(-1)
+    for n in [0, 2, Rational(2, 3), sqrt(2), 3*sqrt(2), 1 + 2*sqrt(3)/5,
+            (2 - 3*sqrt(5))/7, 1 + sqrt(2), (-5 + sqrt(17))/4]:
+        assert (cf_r(cf(n)) - n).expand() == 0
+        assert (cf_r(cf(-n)) + n).expand() == 0
+    raises(ValueError, lambda: cf(sqrt(2 + sqrt(3))))
+    raises(ValueError, lambda: cf(sqrt(2) + sqrt(3)))
+    raises(ValueError, lambda: cf(pi))
+    raises(ValueError, lambda: cf(.1))
+
+    raises(ValueError, lambda: cf_p(1, 0, 0))
+    raises(ValueError, lambda: cf_p(1, 1, -1))
+    assert cf_p(4, 3, 0) == [1, 3]
+    assert cf_p(0, 3, 5) == [0, 1, [2, 1, 12, 1, 2, 2]]
+    assert cf_p(1, 1, 0) == [1]
+    assert cf_p(3, 4, 0) == [0, 1, 3]
+    assert cf_p(4, 5, 0) == [0, 1, 4]
+    assert cf_p(5, 6, 0) == [0, 1, 5]
+    assert cf_p(11, 13, 0) == [0, 1, 5, 2]
+    assert cf_p(16, 19, 0) == [0, 1, 5, 3]
+    assert cf_p(27, 32, 0) == [0, 1, 5, 2, 2]
+    assert cf_p(1, 2, 5) == [[1]]
+    assert cf_p(0, 1, 2) == [1, [2]]
+    assert cf_p(6, 7, 49) == [1, 1, 6]
+    assert cf_p(3796, 1387, 0) == [2, 1, 2, 1, 4]
+    assert cf_p(3245, 10000) == [0, 3, 12, 4, 13]
+    assert cf_p(1932, 2568) == [0, 1, 3, 26, 2]
+    assert cf_p(6589, 2569) == [2, 1, 1, 3, 2, 1, 3, 1, 23]
+
+    def take(iterator, n=7):
+        return list(itertools.islice(iterator, n))
+
+    assert take(cf_i(phi)) == [1, 1, 1, 1, 1, 1, 1]
+    assert take(cf_i(pi)) == [3, 7, 15, 1, 292, 1, 1]
+
+    assert list(cf_i(Rational(17, 12))) == [1, 2, 2, 2]
+    assert list(cf_i(Rational(-17, 12))) == [-2, 1, 1, 2, 2]
+
+    assert list(cf_c([1, 6, 1, 8])) == [S.One, Rational(7, 6), Rational(8, 7), Rational(71, 62)]
+    assert list(cf_c([2])) == [S(2)]
+    assert list(cf_c([1, 1, 1, 1, 1, 1, 1])) == [S.One, S(2), Rational(3, 2), Rational(5, 3),
+                                                 Rational(8, 5), Rational(13, 8), Rational(21, 13)]
+    assert list(cf_c([1, 6, Rational(-1, 2), 4])) == [S.One, Rational(7, 6), Rational(5, 4), Rational(3, 2)]
+    assert take(cf_c([[1]])) == [S.One, S(2), Rational(3, 2), Rational(5, 3), Rational(8, 5),
+                                 Rational(13, 8), Rational(21, 13)]
+    assert take(cf_c([1, [1, 2]])) == [S.One, S(2), Rational(5, 3), Rational(7, 4), Rational(19, 11),
+                                    Rational(26, 15), Rational(71, 41)]
+
+    cf_iter_e = (2 if i == 1 else i // 3 * 2 if i % 3 == 0 else 1 for i in itertools.count(1))
+    assert take(cf_c(cf_iter_e)) == [S(2), S(3), Rational(8, 3), Rational(11, 4), Rational(19, 7),
+                                     Rational(87, 32), Rational(106, 39)]
+
+    assert cf_r([1, 6, 1, 8]) == Rational(71, 62)
+    assert cf_r([3]) == S(3)
+    assert cf_r([-1, 5, 1, 4]) == Rational(-24, 29)
+    assert (cf_r([0, 1, 1, 7, [24, 8]]) - (sqrt(3) + 2)/7).expand() == 0
+    assert cf_r([1, 5, 9]) == Rational(55, 46)
+    assert (cf_r([[1]]) - (sqrt(5) + 1)/2).expand() == 0
+    assert cf_r([-3, 1, 1, [2]]) == -1 - sqrt(2)

rl/Lib/site-packages/sympy/ntheory/tests/test_digits.py

@@ -0,0 +1,55 @@
+from sympy.ntheory import count_digits, digits, is_palindromic
+from sympy.core.intfunc import num_digits
+
+from sympy.testing.pytest import raises
+
+
+def test_num_digits():
+    # depending on whether one rounds up or down or uses log or log10,
+    # one or more of these will fail if you don't check for the off-by
+    # one condition
+    assert num_digits(2, 2) == 2
+    assert num_digits(2**48 - 1, 2) == 48
+    assert num_digits(1000, 10) == 4
+    assert num_digits(125, 5) == 4
+    assert num_digits(100, 16) == 2
+    assert num_digits(-1000, 10) == 4
+    # if changes are made to the function, this structured test over
+    # this range will expose problems
+    for base in range(2, 100):
+        for e in range(1, 100):
+            n = base**e
+            assert num_digits(n, base) == e + 1
+            assert num_digits(n + 1, base) == e + 1
+            assert num_digits(n - 1, base) == e
+
+
+def test_digits():
+    assert all(digits(n, 2)[1:] == [int(d) for d in format(n, 'b')]
+                for n in range(20))
+    assert all(digits(n, 8)[1:] == [int(d) for d in format(n, 'o')]
+                for n in range(20))
+    assert all(digits(n, 16)[1:] == [int(d, 16) for d in format(n, 'x')]
+                for n in range(20))
+    assert digits(2345, 34) == [34, 2, 0, 33]
+    assert digits(384753, 71) == [71, 1, 5, 23, 4]
+    assert digits(93409, 10) == [10, 9, 3, 4, 0, 9]
+    assert digits(-92838, 11) == [-11, 6, 3, 8, 2, 9]
+    assert digits(35, 10) == [10, 3, 5]
+    assert digits(35, 10, 3) == [10, 0, 3, 5]
+    assert digits(-35, 10, 4) == [-10, 0, 0, 3, 5]
+    raises(ValueError, lambda: digits(2, 2, 1))
+
+
+def test_count_digits():
+    assert count_digits(55, 2) == {1: 5, 0: 1}
+    assert count_digits(55, 10) == {5: 2}
+    n = count_digits(123)
+    assert n[4] == 0 and type(n[4]) is int
+
+
+def test_is_palindromic():
+    assert is_palindromic(-11)
+    assert is_palindromic(11)
+    assert is_palindromic(0o121, 8)
+    assert not is_palindromic(123)

rl/Lib/site-packages/sympy/ntheory/tests/test_ecm.py

@@ -0,0 +1,63 @@
+from sympy.external.gmpy import invert
+from sympy.ntheory.ecm import ecm, Point
+from sympy.testing.pytest import slow
+
+@slow
+def test_ecm():
+    assert ecm(3146531246531241245132451321) == {3, 100327907731, 10454157497791297}
+    assert ecm(46167045131415113) == {43, 2634823, 407485517}
+    assert ecm(631211032315670776841) == {9312934919, 67777885039}
+    assert ecm(398883434337287) == {99476569, 4009823}
+    assert ecm(64211816600515193) == {281719, 359641, 633767}
+    assert ecm(4269021180054189416198169786894227) == {184039, 241603, 333331, 477973, 618619, 974123}
+    assert ecm(4516511326451341281684513) == {3, 39869, 131743543, 95542348571}
+    assert ecm(4132846513818654136451) == {47, 160343, 2802377, 195692803}
+    assert ecm(168541512131094651323) == {79, 113, 11011069, 1714635721}
+    #This takes ~10secs while factorint is not able to factorize this even in ~10mins
+    assert ecm(7060005655815754299976961394452809, B1=100000, B2=1000000) == {6988699669998001, 1010203040506070809}
+
+
+def test_Point():
+    #The curve is of the form y**2 = x**3 + a*x**2 + x
+    mod = 101
+    a = 10
+    a_24 = (a + 2)*invert(4, mod)
+    p1 = Point(10, 17, a_24, mod)
+    p2 = p1.double()
+    assert p2 == Point(68, 56, a_24, mod)
+    p4 = p2.double()
+    assert p4 == Point(22, 64, a_24, mod)
+    p8 = p4.double()
+    assert p8 == Point(71, 95, a_24, mod)
+    p16 = p8.double()
+    assert p16 == Point(5, 16, a_24, mod)
+    p32 = p16.double()
+    assert p32 == Point(33, 96, a_24, mod)
+
+    # p3 = p2 + p1
+    p3 = p2.add(p1, p1)
+    assert p3 == Point(1, 61, a_24, mod)
+    # p5 = p3 + p2 or p4 + p1
+    p5 = p3.add(p2, p1)
+    assert p5 == Point(49, 90, a_24, mod)
+    assert p5 == p4.add(p1, p3)
+    # p6 = 2*p3
+    p6 = p3.double()
+    assert p6 == Point(87, 43, a_24, mod)
+    assert p6 == p4.add(p2, p2)
+    # p7 = p5 + p2
+    p7 = p5.add(p2, p3)
+    assert p7 == Point(69, 23, a_24, mod)
+    assert p7 == p4.add(p3, p1)
+    assert p7 == p6.add(p1, p5)
+    # p9 = p5 + p4
+    p9 = p5.add(p4, p1)
+    assert p9 == Point(56, 99, a_24, mod)
+    assert p9 == p6.add(p3, p3)
+    assert p9 == p7.add(p2, p5)
+    assert p9 == p8.add(p1, p7)
+
+    assert p5 == p1.mont_ladder(5)
+    assert p9 == p1.mont_ladder(9)
+    assert p16 == p1.mont_ladder(16)
+    assert p9 == p3.mont_ladder(3)

rl/Lib/site-packages/sympy/ntheory/tests/test_egyptian_fraction.py

@@ -0,0 +1,49 @@
+from sympy.core.numbers import Rational
+from sympy.ntheory.egyptian_fraction import egyptian_fraction
+from sympy.core.add import Add
+from sympy.testing.pytest import raises
+from sympy.core.random import random_complex_number
+
+
+def test_egyptian_fraction():
+    def test_equality(r, alg="Greedy"):
+        return r == Add(*[Rational(1, i) for i in egyptian_fraction(r, alg)])
+
+    r = random_complex_number(a=0, c=1, b=0, d=0, rational=True)
+    assert test_equality(r)
+
+    assert egyptian_fraction(Rational(4, 17)) == [5, 29, 1233, 3039345]
+    assert egyptian_fraction(Rational(7, 13), "Greedy") == [2, 26]
+    assert egyptian_fraction(Rational(23, 101), "Greedy") == \
+        [5, 37, 1438, 2985448, 40108045937720]
+    assert egyptian_fraction(Rational(18, 23), "Takenouchi") == \
+        [2, 6, 12, 35, 276, 2415]
+    assert egyptian_fraction(Rational(5, 6), "Graham Jewett") == \
+        [6, 7, 8, 9, 10, 42, 43, 44, 45, 56, 57, 58, 72, 73, 90, 1806, 1807,
+         1808, 1892, 1893, 1980, 3192, 3193, 3306, 5256, 3263442, 3263443,
+         3267056, 3581556, 10192056, 10650056950806]
+    assert egyptian_fraction(Rational(5, 6), "Golomb") == [2, 6, 12, 20, 30]
+    assert egyptian_fraction(Rational(5, 121), "Golomb") == [25, 1225, 3577, 7081, 11737]
+    raises(ValueError, lambda: egyptian_fraction(Rational(-4, 9)))
+    assert egyptian_fraction(Rational(8, 3), "Golomb") == [1, 2, 3, 4, 5, 6, 7,
+                                                           14, 574, 2788, 6460,
+                                                           11590, 33062, 113820]
+    assert egyptian_fraction(Rational(355, 113)) == [1, 2, 3, 4, 5, 6, 7, 8, 9,
+                                                     10, 11, 12, 27, 744, 893588,
+                                                     1251493536607,
+                                                     20361068938197002344405230]
+
+
+def test_input():
+    r = (2,3), Rational(2, 3), (Rational(2), Rational(3))
+    for m in ["Greedy", "Graham Jewett", "Takenouchi", "Golomb"]:
+        for i in r:
+            d = egyptian_fraction(i, m)
+            assert all(i.is_Integer for i in d)
+            if m == "Graham Jewett":
+                assert d == [3, 4, 12]
+            else:
+                assert d == [2, 6]
+    # check prefix
+    d = egyptian_fraction(Rational(5, 3))
+    assert d == [1, 2, 6] and all(i.is_Integer for i in d)

rl/Lib/site-packages/sympy/ntheory/tests/test_elliptic_curve.py

@@ -0,0 +1,20 @@
+from sympy.ntheory.elliptic_curve import EllipticCurve
+
+
+def test_elliptic_curve():
+    # Point addition and multiplication
+    e3 = EllipticCurve(-1, 9)
+    p = e3(0, 3)
+    q = e3(-1, 3)
+    r = p + q
+    assert r.x == 1 and r.y == -3
+    r = 2*p + q
+    assert r.x == 35 and r.y == 207
+    r = -p + q
+    assert r.x == 37 and r.y == 225
+    # Verify result in http://www.lmfdb.org/EllipticCurve/Q
+    # Discriminant
+    assert EllipticCurve(-1, 9).discriminant == -34928
+    assert EllipticCurve(-2731, -55146, 1, 0, 1).discriminant == 25088
+    # Torsion points
+    assert len(EllipticCurve(0, 1).torsion_points()) == 6

rl/Lib/site-packages/sympy/ntheory/tests/test_factor_.py

@@ -0,0 +1,627 @@
+from sympy.core.containers import Dict
+from sympy.core.mul import Mul
+from sympy.core.power import Pow
+from sympy.core.singleton import S
+from sympy.functions.combinatorial.factorials import factorial as fac
+from sympy.core.numbers import Integer, Rational
+from sympy.external.gmpy import gcd
+
+from sympy.ntheory import (totient,
+    factorint, primefactors, divisors, nextprime,
+    pollard_rho, perfect_power, multiplicity, multiplicity_in_factorial,
+    divisor_count, primorial, pollard_pm1, divisor_sigma,
+    factorrat, reduced_totient)
+from sympy.ntheory.factor_ import (smoothness, smoothness_p, proper_divisors,
+    antidivisors, antidivisor_count, _divisor_sigma, core, udivisors, udivisor_sigma,
+    udivisor_count, proper_divisor_count, primenu, primeomega,
+    mersenne_prime_exponent, is_perfect, is_abundant,
+    is_deficient, is_amicable, is_carmichael, find_carmichael_numbers_in_range,
+    find_first_n_carmichaels, dra, drm, _perfect_power)
+
+from sympy.testing.pytest import raises, slow
+
+from sympy.utilities.iterables import capture
+
+
+def fac_multiplicity(n, p):
+    """Return the power of the prime number p in the
+    factorization of n!"""
+    if p > n:
+        return 0
+    if p > n//2:
+        return 1
+    q, m = n, 0
+    while q >= p:
+        q //= p
+        m += q
+    return m
+
+
+def multiproduct(seq=(), start=1):
+    """
+    Return the product of a sequence of factors with multiplicities,
+    times the value of the parameter ``start``. The input may be a
+    sequence of (factor, exponent) pairs or a dict of such pairs.
+
+        >>> multiproduct({3:7, 2:5}, 4) # = 3**7 * 2**5 * 4
+        279936
+
+    """
+    if not seq:
+        return start
+    if isinstance(seq, dict):
+        seq = iter(seq.items())
+    units = start
+    multi = []
+    for base, exp in seq:
+        if not exp:
+            continue
+        elif exp == 1:
+            units *= base
+        else:
+            if exp % 2:
+                units *= base
+            multi.append((base, exp//2))
+    return units * multiproduct(multi)**2
+
+
+def test_multiplicity():
+    for b in range(2, 20):
+        for i in range(100):
+            assert multiplicity(b, b**i) == i
+            assert multiplicity(b, (b**i) * 23) == i
+            assert multiplicity(b, (b**i) * 1000249) == i
+    # Should be fast
+    assert multiplicity(10, 10**10023) == 10023
+    # Should exit quickly
+    assert multiplicity(10**10, 10**10) == 1
+    # Should raise errors for bad input
+    raises(ValueError, lambda: multiplicity(1, 1))
+    raises(ValueError, lambda: multiplicity(1, 2))
+    raises(ValueError, lambda: multiplicity(1.3, 2))
+    raises(ValueError, lambda: multiplicity(2, 0))
+    raises(ValueError, lambda: multiplicity(1.3, 0))
+
+    # handles Rationals
+    assert multiplicity(10, Rational(30, 7)) == 1
+    assert multiplicity(Rational(2, 7), Rational(4, 7)) == 1
+    assert multiplicity(Rational(1, 7), Rational(3, 49)) == 2
+    assert multiplicity(Rational(2, 7), Rational(7, 2)) == -1
+    assert multiplicity(3, Rational(1, 9)) == -2
+
+
+def test_multiplicity_in_factorial():
+    n = fac(1000)
+    for i in (2, 4, 6, 12, 30, 36, 48, 60, 72, 96):
+        assert multiplicity(i, n) == multiplicity_in_factorial(i, 1000)
+
+
+def test_private_perfect_power():
+    assert _perfect_power(0) is False
+    assert _perfect_power(1) is False
+    assert _perfect_power(2) is False
+    assert _perfect_power(3) is False
+    for x in [2, 3, 5, 6, 7, 12, 15, 105, 100003]:
+        for y in range(2, 100):
+            assert _perfect_power(x**y) == (x, y)
+            if x & 1:
+                assert _perfect_power(x**y, next_p=3) == (x, y)
+            if x == 100003:
+                assert _perfect_power(x**y, next_p=100003) == (x, y)
+            assert _perfect_power(101*x**y) == False
+            # Catalan's conjecture
+            if x**y not in [8, 9]:
+                assert _perfect_power(x**y + 1) == False
+                assert _perfect_power(x**y - 1) == False
+    for x in range(1, 10):
+        for y in range(1, 10):
+            g = gcd(x, y)
+            if g == 1:
+                assert _perfect_power(5**x * 101**y) == False
+            else:
+                assert _perfect_power(5**x * 101**y) == (5**(x//g) * 101**(y//g), g)
+
+
+def test_perfect_power():
+    raises(ValueError, lambda: perfect_power(0.1))
+    assert perfect_power(0) is False
+    assert perfect_power(1) is False
+    assert perfect_power(2) is False
+    assert perfect_power(3) is False
+    assert perfect_power(4) == (2, 2)
+    assert perfect_power(14) is False
+    assert perfect_power(25) == (5, 2)
+    assert perfect_power(22) is False
+    assert perfect_power(22, [2]) is False
+    assert perfect_power(137**(3*5*13)) == (137, 3*5*13)
+    assert perfect_power(137**(3*5*13) + 1) is False
+    assert perfect_power(137**(3*5*13) - 1) is False
+    assert perfect_power(103005006004**7) == (103005006004, 7)
+    assert perfect_power(103005006004**7 + 1) is False
+    assert perfect_power(103005006004**7 - 1) is False
+    assert perfect_power(103005006004**12) == (103005006004, 12)
+    assert perfect_power(103005006004**12 + 1) is False
+    assert perfect_power(103005006004**12 - 1) is False
+    assert perfect_power(2**10007) == (2, 10007)
+    assert perfect_power(2**10007 + 1) is False
+    assert perfect_power(2**10007 - 1) is False
+    assert perfect_power((9**99 + 1)**60) == (9**99 + 1, 60)
+    assert perfect_power((9**99 + 1)**60 + 1) is False
+    assert perfect_power((9**99 + 1)**60 - 1) is False
+    assert perfect_power((10**40000)**2, big=False) == (10**40000, 2)
+    assert perfect_power(10**100000) == (10, 100000)
+    assert perfect_power(10**100001) == (10, 100001)
+    assert perfect_power(13**4, [3, 5]) is False
+    assert perfect_power(3**4, [3, 10], factor=0) is False
+    assert perfect_power(3**3*5**3) == (15, 3)
+    assert perfect_power(2**3*5**5) is False
+    assert perfect_power(2*13**4) is False
+    assert perfect_power(2**5*3**3) is False
+    t = 2**24
+    for d in divisors(24):
+        m = perfect_power(t*3**d)
+        assert m and m[1] == d or d == 1
+        m = perfect_power(t*3**d, big=False)
+        assert m and m[1] == 2 or d == 1 or d == 3, (d, m)
+
+    # negatives and non-integer rationals
+    assert perfect_power(-4) is False
+    assert perfect_power(-8) == (-2, 3)
+    assert perfect_power(Rational(1, 2)**3) == (S.Half, 3)
+    assert perfect_power(Rational(-3, 2)**3) == (-3*S.Half, 3)
+
+
+@slow
+def test_factorint():
+    assert primefactors(123456) == [2, 3, 643]
+    assert factorint(0) == {0: 1}
+    assert factorint(1) == {}
+    assert factorint(-1) == {-1: 1}
+    assert factorint(-2) == {-1: 1, 2: 1}
+    assert factorint(-16) == {-1: 1, 2: 4}
+    assert factorint(2) == {2: 1}
+    assert factorint(126) == {2: 1, 3: 2, 7: 1}
+    assert factorint(123456) == {2: 6, 3: 1, 643: 1}
+    assert factorint(5951757) == {3: 1, 7: 1, 29: 2, 337: 1}
+    assert factorint(64015937) == {7993: 1, 8009: 1}
+    assert factorint(2**(2**6) + 1) == {274177: 1, 67280421310721: 1}
+    #issue 19683
+    assert factorint(10**38 - 1) == {3: 2, 11: 1, 909090909090909091: 1, 1111111111111111111: 1}
+    #issue 17676
+    assert factorint(28300421052393658575) == {3: 1, 5: 2, 11: 2, 43: 1, 2063: 2, 4127: 1, 4129: 1}
+    assert factorint(2063**2 * 4127**1 * 4129**1) == {2063: 2, 4127: 1, 4129: 1}
+    assert factorint(2347**2 * 7039**1 * 7043**1) == {2347: 2, 7039: 1, 7043: 1}
+
+    assert factorint(0, multiple=True) == [0]
+    assert factorint(1, multiple=True) == []
+    assert factorint(-1, multiple=True) == [-1]
+    assert factorint(-2, multiple=True) == [-1, 2]
+    assert factorint(-16, multiple=True) == [-1, 2, 2, 2, 2]
+    assert factorint(2, multiple=True) == [2]
+    assert factorint(24, multiple=True) == [2, 2, 2, 3]
+    assert factorint(126, multiple=True) == [2, 3, 3, 7]
+    assert factorint(123456, multiple=True) == [2, 2, 2, 2, 2, 2, 3, 643]
+    assert factorint(5951757, multiple=True) == [3, 7, 29, 29, 337]
+    assert factorint(64015937, multiple=True) == [7993, 8009]
+    assert factorint(2**(2**6) + 1, multiple=True) == [274177, 67280421310721]
+
+    assert factorint(fac(1, evaluate=False)) == {}
+    assert factorint(fac(7, evaluate=False)) == {2: 4, 3: 2, 5: 1, 7: 1}
+    assert factorint(fac(15, evaluate=False)) == \
+        {2: 11, 3: 6, 5: 3, 7: 2, 11: 1, 13: 1}
+    assert factorint(fac(20, evaluate=False)) == \
+        {2: 18, 3: 8, 5: 4, 7: 2, 11: 1, 13: 1, 17: 1, 19: 1}
+    assert factorint(fac(23, evaluate=False)) == \
+        {2: 19, 3: 9, 5: 4, 7: 3, 11: 2, 13: 1, 17: 1, 19: 1, 23: 1}
+
+    assert multiproduct(factorint(fac(200))) == fac(200)
+    assert multiproduct(factorint(fac(200, evaluate=False))) == fac(200)
+    for b, e in factorint(fac(150)).items():
+        assert e == fac_multiplicity(150, b)
+    for b, e in factorint(fac(150, evaluate=False)).items():
+        assert e == fac_multiplicity(150, b)
+    assert factorint(103005006059**7) == {103005006059: 7}
+    assert factorint(31337**191) == {31337: 191}
+    assert factorint(2**1000 * 3**500 * 257**127 * 383**60) == \
+        {2: 1000, 3: 500, 257: 127, 383: 60}
+    assert len(factorint(fac(10000))) == 1229
+    assert len(factorint(fac(10000, evaluate=False))) == 1229
+    assert factorint(12932983746293756928584532764589230) == \
+        {2: 1, 5: 1, 73: 1, 727719592270351: 1, 63564265087747: 1, 383: 1}
+    assert factorint(727719592270351) == {727719592270351: 1}
+    assert factorint(2**64 + 1, use_trial=False) == factorint(2**64 + 1)
+    for n in range(60000):
+        assert multiproduct(factorint(n)) == n
+    assert pollard_rho(2**64 + 1, seed=1) == 274177
+    assert pollard_rho(19, seed=1) is None
+    assert factorint(3, limit=2) == {3: 1}
+    assert factorint(12345) == {3: 1, 5: 1, 823: 1}
+    assert factorint(
+        12345, limit=3) == {4115: 1, 3: 1}  # the 5 is greater than the limit
+    assert factorint(1, limit=1) == {}
+    assert factorint(0, 3) == {0: 1}
+    assert factorint(12, limit=1) == {12: 1}
+    assert factorint(30, limit=2) == {2: 1, 15: 1}
+    assert factorint(16, limit=2) == {2: 4}
+    assert factorint(124, limit=3) == {2: 2, 31: 1}
+    assert factorint(4*31**2, limit=3) == {2: 2, 31: 2}
+    p1 = nextprime(2**32)
+    p2 = nextprime(2**16)
+    p3 = nextprime(p2)
+    assert factorint(p1*p2*p3) == {p1: 1, p2: 1, p3: 1}
+    assert factorint(13*17*19, limit=15) == {13: 1, 17*19: 1}
+    assert factorint(1951*15013*15053, limit=2000) == {225990689: 1, 1951: 1}
+    assert factorint(primorial(17) + 1, use_pm1=0) == \
+        {int(19026377261): 1, 3467: 1, 277: 1, 105229: 1}
+    # when prime b is closer than approx sqrt(8*p) to prime p then they are
+    # "close" and have a trivial factorization
+    a = nextprime(2**2**8)  # 78 digits
+    b = nextprime(a + 2**2**4)
+    assert 'Fermat' in capture(lambda: factorint(a*b, verbose=1))
+
+    raises(ValueError, lambda: pollard_rho(4))
+    raises(ValueError, lambda: pollard_pm1(3))
+    raises(ValueError, lambda: pollard_pm1(10, B=2))
+    # verbose coverage
+    n = nextprime(2**16)*nextprime(2**17)*nextprime(1901)
+    assert 'with primes' in capture(lambda: factorint(n, verbose=1))
+    capture(lambda: factorint(nextprime(2**16)*1012, verbose=1))
+
+    n = nextprime(2**17)
+    capture(lambda: factorint(n**3, verbose=1))  # perfect power termination
+    capture(lambda: factorint(2*n, verbose=1))  # factoring complete msg
+
+    # exceed 1st
+    n = nextprime(2**17)
+    n *= nextprime(n)
+    assert '1000' in capture(lambda: factorint(n, limit=1000, verbose=1))
+    n *= nextprime(n)
+    assert len(factorint(n)) == 3
+    assert len(factorint(n, limit=p1)) == 3
+    n *= nextprime(2*n)
+    # exceed 2nd
+    assert '2001' in capture(lambda: factorint(n, limit=2000, verbose=1))
+    assert capture(
+        lambda: factorint(n, limit=4000, verbose=1)).count('Pollard') == 2
+    # non-prime pm1 result
+    n = nextprime(8069)
+    n *= nextprime(2*n)*nextprime(2*n, 2)
+    capture(lambda: factorint(n, verbose=1))  # non-prime pm1 result
+    # factor fermat composite
+    p1 = nextprime(2**17)
+    p2 = nextprime(2*p1)
+    assert factorint((p1*p2**2)**3) == {p1: 3, p2: 6}
+    # Test for non integer input
+    raises(ValueError, lambda: factorint(4.5))
+    # test dict/Dict input
+    sans = '2**10*3**3'
+    n = {4: 2, 12: 3}
+    assert str(factorint(n)) == sans
+    assert str(factorint(Dict(n))) == sans
+
+
+def test_divisors_and_divisor_count():
+    assert divisors(-1) == [1]
+    assert divisors(0) == []
+    assert divisors(1) == [1]
+    assert divisors(2) == [1, 2]
+    assert divisors(3) == [1, 3]
+    assert divisors(17) == [1, 17]
+    assert divisors(10) == [1, 2, 5, 10]
+    assert divisors(100) == [1, 2, 4, 5, 10, 20, 25, 50, 100]
+    assert divisors(101) == [1, 101]
+    assert type(divisors(2, generator=True)) is not list
+
+    assert divisor_count(0) == 0
+    assert divisor_count(-1) == 1
+    assert divisor_count(1) == 1
+    assert divisor_count(6) == 4
+    assert divisor_count(12) == 6
+
+    assert divisor_count(180, 3) == divisor_count(180//3)
+    assert divisor_count(2*3*5, 7) == 0
+
+
+def test_proper_divisors_and_proper_divisor_count():
+    assert proper_divisors(-1) == []
+    assert proper_divisors(0) == []
+    assert proper_divisors(1) == []
+    assert proper_divisors(2) == [1]
+    assert proper_divisors(3) == [1]
+    assert proper_divisors(17) == [1]
+    assert proper_divisors(10) == [1, 2, 5]
+    assert proper_divisors(100) == [1, 2, 4, 5, 10, 20, 25, 50]
+    assert proper_divisors(1000000007) == [1]
+    assert type(proper_divisors(2, generator=True)) is not list
+
+    assert proper_divisor_count(0) == 0
+    assert proper_divisor_count(-1) == 0
+    assert proper_divisor_count(1) == 0
+    assert proper_divisor_count(36) == 8
+    assert proper_divisor_count(2*3*5) == 7
+
+
+def test_udivisors_and_udivisor_count():
+    assert udivisors(-1) == [1]
+    assert udivisors(0) == []
+    assert udivisors(1) == [1]
+    assert udivisors(2) == [1, 2]
+    assert udivisors(3) == [1, 3]
+    assert udivisors(17) == [1, 17]
+    assert udivisors(10) == [1, 2, 5, 10]
+    assert udivisors(100) == [1, 4, 25, 100]
+    assert udivisors(101) == [1, 101]
+    assert udivisors(1000) == [1, 8, 125, 1000]
+    assert type(udivisors(2, generator=True)) is not list
+
+    assert udivisor_count(0) == 0
+    assert udivisor_count(-1) == 1
+    assert udivisor_count(1) == 1
+    assert udivisor_count(6) == 4
+    assert udivisor_count(12) == 4
+
+    assert udivisor_count(180) == 8
+    assert udivisor_count(2*3*5*7) == 16
+
+
+def test_issue_6981():
+    S = set(divisors(4)).union(set(divisors(Integer(2))))
+    assert S == {1,2,4}
+
+
+def test_issue_4356():
+    assert factorint(1030903) == {53: 2, 367: 1}
+
+
+def test_divisors():
+    assert divisors(28) == [1, 2, 4, 7, 14, 28]
+    assert list(divisors(3*5*7, 1)) == [1, 3, 5, 15, 7, 21, 35, 105]
+    assert divisors(0) == []
+
+
+def test_divisor_count():
+    assert divisor_count(0) == 0
+    assert divisor_count(6) == 4
+
+
+def test_proper_divisors():
+    assert proper_divisors(-1) == []
+    assert proper_divisors(28) == [1, 2, 4, 7, 14]
+    assert list(proper_divisors(3*5*7, True)) == [1, 3, 5, 15, 7, 21, 35]
+
+
+def test_proper_divisor_count():
+    assert proper_divisor_count(6) == 3
+    assert proper_divisor_count(108) == 11
+
+
+def test_antidivisors():
+    assert antidivisors(-1) == []
+    assert antidivisors(-3) == [2]
+    assert antidivisors(14) == [3, 4, 9]
+    assert antidivisors(237) == [2, 5, 6, 11, 19, 25, 43, 95, 158]
+    assert antidivisors(12345) == [2, 6, 7, 10, 30, 1646, 3527, 4938, 8230]
+    assert antidivisors(393216) == [262144]
+    assert sorted(x for x in antidivisors(3*5*7, 1)) == \
+        [2, 6, 10, 11, 14, 19, 30, 42, 70]
+    assert antidivisors(1) == []
+    assert type(antidivisors(2, generator=True)) is not list
+
+def test_antidivisor_count():
+    assert antidivisor_count(0) == 0
+    assert antidivisor_count(-1) == 0
+    assert antidivisor_count(-4) == 1
+    assert antidivisor_count(20) == 3
+    assert antidivisor_count(25) == 5
+    assert antidivisor_count(38) == 7
+    assert antidivisor_count(180) == 6
+    assert antidivisor_count(2*3*5) == 3
+
+
+def test_smoothness_and_smoothness_p():
+    assert smoothness(1) == (1, 1)
+    assert smoothness(2**4*3**2) == (3, 16)
+
+    assert smoothness_p(10431, m=1) == \
+        (1, [(3, (2, 2, 4)), (19, (1, 5, 5)), (61, (1, 31, 31))])
+    assert smoothness_p(10431) == \
+        (-1, [(3, (2, 2, 2)), (19, (1, 3, 9)), (61, (1, 5, 5))])
+    assert smoothness_p(10431, power=1) == \
+        (-1, [(3, (2, 2, 2)), (61, (1, 5, 5)), (19, (1, 3, 9))])
+    assert smoothness_p(21477639576571, visual=1) == \
+        'p**i=4410317**1 has p-1 B=1787, B-pow=1787\n' + \
+        'p**i=4869863**1 has p-1 B=2434931, B-pow=2434931'
+
+
+def test_visual_factorint():
+    assert factorint(1, visual=1) == 1
+    forty2 = factorint(42, visual=True)
+    assert type(forty2) == Mul
+    assert str(forty2) == '2**1*3**1*7**1'
+    assert factorint(1, visual=True) is S.One
+    no = {"evaluate": False}
+    assert factorint(42**2, visual=True) == Mul(Pow(2, 2, **no),
+                                                Pow(3, 2, **no),
+                                                Pow(7, 2, **no), **no)
+    assert -1 in factorint(-42, visual=True).args
+
+
+def test_factorrat():
+    assert str(factorrat(S(12)/1, visual=True)) == '2**2*3**1'
+    assert str(factorrat(Rational(1, 1), visual=True)) == '1'
+    assert str(factorrat(S(25)/14, visual=True)) == '5**2/(2*7)'
+    assert str(factorrat(Rational(25, 14), visual=True)) == '5**2/(2*7)'
+    assert str(factorrat(S(-25)/14/9, visual=True)) == '-1*5**2/(2*3**2*7)'
+
+    assert factorrat(S(12)/1, multiple=True) == [2, 2, 3]
+    assert factorrat(Rational(1, 1), multiple=True) == []
+    assert factorrat(S(25)/14, multiple=True) == [Rational(1, 7), S.Half, 5, 5]
+    assert factorrat(Rational(25, 14), multiple=True) == [Rational(1, 7), S.Half, 5, 5]
+    assert factorrat(Rational(12, 1), multiple=True) == [2, 2, 3]
+    assert factorrat(S(-25)/14/9, multiple=True) == \
+        [-1, Rational(1, 7), Rational(1, 3), Rational(1, 3), S.Half, 5, 5]
+
+
+def test_visual_io():
+    sm = smoothness_p
+    fi = factorint
+    # with smoothness_p
+    n = 124
+    d = fi(n)
+    m = fi(d, visual=True)
+    t = sm(n)
+    s = sm(t)
+    for th in [d, s, t, n, m]:
+        assert sm(th, visual=True) == s
+        assert sm(th, visual=1) == s
+    for th in [d, s, t, n, m]:
+        assert sm(th, visual=False) == t
+    assert [sm(th, visual=None) for th in [d, s, t, n, m]] == [s, d, s, t, t]
+    assert [sm(th, visual=2) for th in [d, s, t, n, m]] == [s, d, s, t, t]
+
+    # with factorint
+    for th in [d, m, n]:
+        assert fi(th, visual=True) == m
+        assert fi(th, visual=1) == m
+    for th in [d, m, n]:
+        assert fi(th, visual=False) == d
+    assert [fi(th, visual=None) for th in [d, m, n]] == [m, d, d]
+    assert [fi(th, visual=0) for th in [d, m, n]] == [m, d, d]
+
+    # test reevaluation
+    no = {"evaluate": False}
+    assert sm({4: 2}, visual=False) == sm(16)
+    assert sm(Mul(*[Pow(k, v, **no) for k, v in {4: 2, 2: 6}.items()], **no),
+              visual=False) == sm(2**10)
+
+    assert fi({4: 2}, visual=False) == fi(16)
+    assert fi(Mul(*[Pow(k, v, **no) for k, v in {4: 2, 2: 6}.items()], **no),
+              visual=False) == fi(2**10)
+
+
+def test_core():
+    assert core(35**13, 10) == 42875
+    assert core(210**2) == 1
+    assert core(7776, 3) == 36
+    assert core(10**27, 22) == 10**5
+    assert core(537824) == 14
+    assert core(1, 6) == 1
+
+
+def test__divisor_sigma():
+    assert _divisor_sigma(23450) == 50592
+    assert _divisor_sigma(23450, 0) == 24
+    assert _divisor_sigma(23450, 1) == 50592
+    assert _divisor_sigma(23450, 2) == 730747500
+    assert _divisor_sigma(23450, 3) == 14666785333344
+    A000005 = [1, 2, 2, 3, 2, 4, 2, 4, 3, 4, 2, 6, 2, 4, 4, 5, 2, 6, 2, 6, 4,
+               4, 2, 8, 3, 4, 4, 6, 2, 8, 2, 6, 4, 4, 4, 9, 2, 4, 4, 8, 2, 8]
+    for n, val in enumerate(A000005, 1):
+        assert _divisor_sigma(n, 0) == val
+    A000203 = [1, 3, 4, 7, 6, 12, 8, 15, 13, 18, 12, 28, 14, 24, 24, 31, 18,
+               39, 20, 42, 32, 36, 24, 60, 31, 42, 40, 56, 30, 72, 32, 63, 48]
+    for n, val in enumerate(A000203, 1):
+        assert _divisor_sigma(n, 1) == val
+    A001157 = [1, 5, 10, 21, 26, 50, 50, 85, 91, 130, 122, 210, 170, 250, 260,
+               341, 290, 455, 362, 546, 500, 610, 530, 850, 651, 850, 820, 1050]
+    for n, val in enumerate(A001157, 1):
+        assert _divisor_sigma(n, 2) == val
+
+
+def test_mersenne_prime_exponent():
+    assert mersenne_prime_exponent(1) == 2
+    assert mersenne_prime_exponent(4) == 7
+    assert mersenne_prime_exponent(10) == 89
+    assert mersenne_prime_exponent(25) == 21701
+    raises(ValueError, lambda: mersenne_prime_exponent(52))
+    raises(ValueError, lambda: mersenne_prime_exponent(0))
+
+
+def test_is_perfect():
+    assert is_perfect(-6) is False
+    assert is_perfect(6) is True
+    assert is_perfect(15) is False
+    assert is_perfect(28) is True
+    assert is_perfect(400) is False
+    assert is_perfect(496) is True
+    assert is_perfect(8128) is True
+    assert is_perfect(10000) is False
+
+
+def test_is_abundant():
+    assert is_abundant(10) is False
+    assert is_abundant(12) is True
+    assert is_abundant(18) is True
+    assert is_abundant(21) is False
+    assert is_abundant(945) is True
+
+
+def test_is_deficient():
+    assert is_deficient(10) is True
+    assert is_deficient(22) is True
+    assert is_deficient(56) is False
+    assert is_deficient(20) is False
+    assert is_deficient(36) is False
+
+
+def test_is_amicable():
+    assert is_amicable(173, 129) is False
+    assert is_amicable(220, 284) is True
+    assert is_amicable(8756, 8756) is False
+
+
+def test_is_carmichael():
+    A002997 = [561, 1105, 1729, 2465, 2821, 6601, 8911, 10585, 15841,
+               29341, 41041, 46657, 52633, 62745, 63973, 75361, 101101]
+    for n in range(1, 5000):
+        assert is_carmichael(n) == (n in A002997)
+    for n in A002997:
+        assert is_carmichael(n)
+
+
+def test_find_carmichael_numbers_in_range():
+    assert find_carmichael_numbers_in_range(0, 561) == []
+    assert find_carmichael_numbers_in_range(561, 562) == [561]
+    assert find_carmichael_numbers_in_range(561, 1105) == find_carmichael_numbers_in_range(561, 562)
+    raises(ValueError, lambda: find_carmichael_numbers_in_range(-2, 2))
+    raises(ValueError, lambda: find_carmichael_numbers_in_range(22, 2))
+
+
+def test_find_first_n_carmichaels():
+    assert find_first_n_carmichaels(0) == []
+    assert find_first_n_carmichaels(1) == [561]
+    assert find_first_n_carmichaels(2) == [561, 1105]
+
+
+def test_dra():
+    assert dra(19, 12) == 8
+    assert dra(2718, 10) == 9
+    assert dra(0, 22) == 0
+    assert dra(23456789, 10) == 8
+    raises(ValueError, lambda: dra(24, -2))
+    raises(ValueError, lambda: dra(24.2, 5))
+
+def test_drm():
+    assert drm(19, 12) == 7
+    assert drm(2718, 10) == 2
+    assert drm(0, 15) == 0
+    assert drm(234161, 10) == 6
+    raises(ValueError, lambda: drm(24, -2))
+    raises(ValueError, lambda: drm(11.6, 9))
+
+
+def test_deprecated_ntheory_symbolic_functions():
+    from sympy.testing.pytest import warns_deprecated_sympy
+
+    with warns_deprecated_sympy():
+        assert primenu(3) == 1
+    with warns_deprecated_sympy():
+        assert primeomega(3) == 1
+    with warns_deprecated_sympy():
+        assert totient(3) == 2
+    with warns_deprecated_sympy():
+        assert reduced_totient(3) == 2
+    with warns_deprecated_sympy():
+        assert divisor_sigma(3) == 4
+    with warns_deprecated_sympy():
+        assert udivisor_sigma(3) == 4

rl/Lib/site-packages/sympy/ntheory/tests/test_generate.py

@@ -0,0 +1,285 @@
+from bisect import bisect, bisect_left
+
+from sympy.functions.combinatorial.numbers import mobius, totient
+from sympy.ntheory.generate import (sieve, Sieve)
+
+from sympy.ntheory import isprime, randprime, nextprime, prevprime, \
+    primerange, primepi, prime, primorial, composite, compositepi
+from sympy.ntheory.generate import cycle_length, _primepi
+from sympy.ntheory.primetest import mr
+from sympy.testing.pytest import raises
+
+def test_prime():
+    assert prime(1) == 2
+    assert prime(2) == 3
+    assert prime(5) == 11
+    assert prime(11) == 31
+    assert prime(57) == 269
+    assert prime(296) == 1949
+    assert prime(559) == 4051
+    assert prime(3000) == 27449
+    assert prime(4096) == 38873
+    assert prime(9096) == 94321
+    assert prime(25023) == 287341
+    assert prime(10000000) == 179424673 # issue #20951
+    assert prime(99999999) == 2038074739
+    raises(ValueError, lambda: prime(0))
+    sieve.extend(3000)
+    assert prime(401) == 2749
+    raises(ValueError, lambda: prime(-1))
+
+
+def test__primepi():
+    assert _primepi(-1) == 0
+    assert _primepi(1) == 0
+    assert _primepi(2) == 1
+    assert _primepi(5) == 3
+    assert _primepi(11) == 5
+    assert _primepi(57) == 16
+    assert _primepi(296) == 62
+    assert _primepi(559) == 102
+    assert _primepi(3000) == 430
+    assert _primepi(4096) == 564
+    assert _primepi(9096) == 1128
+    assert _primepi(25023) == 2763
+    assert _primepi(10**8) == 5761455
+    assert _primepi(253425253) == 13856396
+    assert _primepi(8769575643) == 401464322
+    sieve.extend(3000)
+    assert _primepi(2000) == 303
+
+
+def test_composite():
+    from sympy.ntheory.generate import sieve
+    sieve._reset()
+    assert composite(1) == 4
+    assert composite(2) == 6
+    assert composite(5) == 10
+    assert composite(11) == 20
+    assert composite(41) == 58
+    assert composite(57) == 80
+    assert composite(296) == 370
+    assert composite(559) == 684
+    assert composite(3000) == 3488
+    assert composite(4096) == 4736
+    assert composite(9096) == 10368
+    assert composite(25023) == 28088
+    sieve.extend(3000)
+    assert composite(1957) == 2300
+    assert composite(2568) == 2998
+    raises(ValueError, lambda: composite(0))
+
+
+def test_compositepi():
+    assert compositepi(1) == 0
+    assert compositepi(2) == 0
+    assert compositepi(5) == 1
+    assert compositepi(11) == 5
+    assert compositepi(57) == 40
+    assert compositepi(296) == 233
+    assert compositepi(559) == 456
+    assert compositepi(3000) == 2569
+    assert compositepi(4096) == 3531
+    assert compositepi(9096) == 7967
+    assert compositepi(25023) == 22259
+    assert compositepi(10**8) == 94238544
+    assert compositepi(253425253) == 239568856
+    assert compositepi(8769575643) == 8368111320
+    sieve.extend(3000)
+    assert compositepi(2321) == 1976
+
+
+def test_generate():
+    from sympy.ntheory.generate import sieve
+    sieve._reset()
+    assert nextprime(-4) == 2
+    assert nextprime(2) == 3
+    assert nextprime(5) == 7
+    assert nextprime(12) == 13
+    assert prevprime(3) == 2
+    assert prevprime(7) == 5
+    assert prevprime(13) == 11
+    assert prevprime(19) == 17
+    assert prevprime(20) == 19
+
+    sieve.extend_to_no(9)
+    assert sieve._list[-1] == 23
+
+    assert sieve._list[-1] < 31
+    assert 31 in sieve
+
+    assert nextprime(90) == 97
+    assert nextprime(10**40) == (10**40 + 121)
+    primelist = [2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31,
+                 37, 41, 43, 47, 53, 59, 61, 67, 71, 73,
+                 79, 83, 89, 97, 101, 103, 107, 109, 113,
+                 127, 131, 137, 139, 149, 151, 157, 163,
+                 167, 173, 179, 181, 191, 193, 197, 199,
+                 211, 223, 227, 229, 233, 239, 241, 251,
+                 257, 263, 269, 271, 277, 281, 283, 293]
+    for i in range(len(primelist) - 2):
+        for j in range(2, len(primelist) - i):
+            assert nextprime(primelist[i], j) == primelist[i + j]
+            if 3 < i:
+                assert nextprime(primelist[i] - 1, j) == primelist[i + j - 1]
+    raises(ValueError, lambda: nextprime(2, 0))
+    raises(ValueError, lambda: nextprime(2, -1))
+    assert prevprime(97) == 89
+    assert prevprime(10**40) == (10**40 - 17)
+
+    raises(ValueError, lambda: Sieve(0))
+    raises(ValueError, lambda: Sieve(-1))
+    for sieve_interval in [1, 10, 11, 1_000_000]:
+        s = Sieve(sieve_interval=sieve_interval)
+        for head in range(s._list[-1] + 1, (s._list[-1] + 1)**2, 2):
+            for tail in range(head + 1, (s._list[-1] + 1)**2):
+                A = list(s._primerange(head, tail))
+                B = primelist[bisect(primelist, head):bisect_left(primelist, tail)]
+                assert A == B
+        for k in range(s._list[-1], primelist[-1] - 1, 2):
+            s = Sieve(sieve_interval=sieve_interval)
+            s.extend(k)
+            assert list(s._list) == primelist[:bisect(primelist, k)]
+            s.extend(primelist[-1])
+            assert list(s._list) == primelist
+
+    assert list(sieve.primerange(10, 1)) == []
+    assert list(sieve.primerange(5, 9)) == [5, 7]
+    sieve._reset(prime=True)
+    assert list(sieve.primerange(2, 13)) == [2, 3, 5, 7, 11]
+    assert list(sieve.primerange(13)) == [2, 3, 5, 7, 11]
+    assert list(sieve.primerange(8)) == [2, 3, 5, 7]
+    assert list(sieve.primerange(-2)) == []
+    assert list(sieve.primerange(29)) == [2, 3, 5, 7, 11, 13, 17, 19, 23]
+    assert list(sieve.primerange(34)) == [2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31]
+
+    assert list(sieve.totientrange(5, 15)) == [4, 2, 6, 4, 6, 4, 10, 4, 12, 6]
+    sieve._reset(totient=True)
+    assert list(sieve.totientrange(3, 13)) == [2, 2, 4, 2, 6, 4, 6, 4, 10, 4]
+    assert list(sieve.totientrange(900, 1000)) == [totient(x) for x in range(900, 1000)]
+    assert list(sieve.totientrange(0, 1)) == []
+    assert list(sieve.totientrange(1, 2)) == [1]
+
+    assert list(sieve.mobiusrange(5, 15)) == [-1, 1, -1, 0, 0, 1, -1, 0, -1, 1]
+    sieve._reset(mobius=True)
+    assert list(sieve.mobiusrange(3, 13)) == [-1, 0, -1, 1, -1, 0, 0, 1, -1, 0]
+    assert list(sieve.mobiusrange(1050, 1100)) == [mobius(x) for x in range(1050, 1100)]
+    assert list(sieve.mobiusrange(0, 1)) == []
+    assert list(sieve.mobiusrange(1, 2)) == [1]
+
+    assert list(primerange(10, 1)) == []
+    assert list(primerange(2, 7)) == [2, 3, 5]
+    assert list(primerange(2, 10)) == [2, 3, 5, 7]
+    assert list(primerange(1050, 1100)) == [1051, 1061,
+        1063, 1069, 1087, 1091, 1093, 1097]
+    s = Sieve()
+    for i in range(30, 2350, 376):
+        for j in range(2, 5096, 1139):
+            A = list(s.primerange(i, i + j))
+            B = list(primerange(i, i + j))
+            assert A == B
+    s = Sieve()
+    sieve._reset(prime=True)
+    sieve.extend(13)
+    for i in range(200):
+        for j in range(i, 200):
+            A = list(s.primerange(i, j))
+            B = list(primerange(i, j))
+            assert A == B
+    sieve.extend(1000)
+    for a, b in [(901, 1103), # a < 1000 < b < 1000**2
+                 (806, 1002007), # a < 1000 < 1000**2 < b
+                 (2000, 30001), # 1000 < a < b < 1000**2
+                 (100005, 1010001), # 1000 < a < 1000**2 < b
+                 (1003003, 1005000), # 1000**2 < a < b
+                 ]:
+        assert list(primerange(a, b)) == list(s.primerange(a, b))
+    sieve._reset(prime=True)
+    sieve.extend(100000)
+    assert len(sieve._list) == len(set(sieve._list))
+    s = Sieve()
+    assert s[10] == 29
+
+    assert nextprime(2, 2) == 5
+
+    raises(ValueError, lambda: totient(0))
+
+    raises(ValueError, lambda: primorial(0))
+
+    assert mr(1, [2]) is False
+
+    func = lambda i: (i**2 + 1) % 51
+    assert next(cycle_length(func, 4)) == (6, 3)
+    assert list(cycle_length(func, 4, values=True)) == \
+        [4, 17, 35, 2, 5, 26, 14, 44, 50, 2, 5, 26, 14]
+    assert next(cycle_length(func, 4, nmax=5)) == (5, None)
+    assert list(cycle_length(func, 4, nmax=5, values=True)) == \
+        [4, 17, 35, 2, 5]
+    sieve.extend(3000)
+    assert nextprime(2968) == 2969
+    assert prevprime(2930) == 2927
+    raises(ValueError, lambda: prevprime(1))
+    raises(ValueError, lambda: prevprime(-4))
+
+
+def test_randprime():
+    assert randprime(10, 1) is None
+    assert randprime(3, -3) is None
+    assert randprime(2, 3) == 2
+    assert randprime(1, 3) == 2
+    assert randprime(3, 5) == 3
+    raises(ValueError, lambda: randprime(-12, -2))
+    raises(ValueError, lambda: randprime(-10, 0))
+    raises(ValueError, lambda: randprime(20, 22))
+    raises(ValueError, lambda: randprime(0, 2))
+    raises(ValueError, lambda: randprime(1, 2))
+    for a in [100, 300, 500, 250000]:
+        for b in [100, 300, 500, 250000]:
+            p = randprime(a, a + b)
+            assert a <= p < (a + b) and isprime(p)
+
+
+def test_primorial():
+    assert primorial(1) == 2
+    assert primorial(1, nth=0) == 1
+    assert primorial(2) == 6
+    assert primorial(2, nth=0) == 2
+    assert primorial(4, nth=0) == 6
+
+
+def test_search():
+    assert 2 in sieve
+    assert 2.1 not in sieve
+    assert 1 not in sieve
+    assert 2**1000 not in sieve
+    raises(ValueError, lambda: sieve.search(1))
+
+
+def test_sieve_slice():
+    assert sieve[5] == 11
+    assert list(sieve[5:10]) == [sieve[x] for x in range(5, 10)]
+    assert list(sieve[5:10:2]) == [sieve[x] for x in range(5, 10, 2)]
+    assert list(sieve[1:5]) == [2, 3, 5, 7]
+    raises(IndexError, lambda: sieve[:5])
+    raises(IndexError, lambda: sieve[0])
+    raises(IndexError, lambda: sieve[0:5])
+
+def test_sieve_iter():
+    values = []
+    for value in sieve:
+        if value > 7:
+            break
+        values.append(value)
+    assert values == list(sieve[1:5])
+
+
+def test_sieve_repr():
+    assert "sieve" in repr(sieve)
+    assert "prime" in repr(sieve)
+
+
+def test_deprecated_ntheory_symbolic_functions():
+    from sympy.testing.pytest import warns_deprecated_sympy
+
+    with warns_deprecated_sympy():
+        assert primepi(0) == 0

rl/Lib/site-packages/sympy/ntheory/tests/test_hypothesis.py

@@ -0,0 +1,24 @@
+from hypothesis import given
+from hypothesis import strategies as st
+from sympy import divisors
+from sympy.functions.combinatorial.numbers import divisor_sigma, totient
+from sympy.ntheory.primetest import is_square
+
+
+@given(n=st.integers(1, 10**10))
+def test_tau_hypothesis(n):
+    div = divisors(n)
+    tau_n = len(div)
+    assert is_square(n) == (tau_n % 2 == 1)
+    sigmas = [divisor_sigma(i) for i in div]
+    totients = [totient(n // i) for i in div]
+    mul = [a * b for a, b in zip(sigmas, totients)]
+    assert n * tau_n == sum(mul)
+
+
+@given(n=st.integers(1, 10**10))
+def test_totient_hypothesis(n):
+    assert totient(n) <= n
+    div = divisors(n)
+    totients = [totient(i) for i in div]
+    assert n == sum(totients)

rl/Lib/site-packages/sympy/ntheory/tests/test_modular.py

@@ -0,0 +1,34 @@
+from sympy.ntheory.modular import crt, crt1, crt2, solve_congruence
+from sympy.testing.pytest import raises
+
+
+def test_crt():
+    def mcrt(m, v, r, symmetric=False):
+        assert crt(m, v, symmetric)[0] == r
+        mm, e, s = crt1(m)
+        assert crt2(m, v, mm, e, s, symmetric) == (r, mm)
+
+    mcrt([2, 3, 5], [0, 0, 0], 0)
+    mcrt([2, 3, 5], [1, 1, 1], 1)
+
+    mcrt([2, 3, 5], [-1, -1, -1], -1, True)
+    mcrt([2, 3, 5], [-1, -1, -1], 2*3*5 - 1, False)
+
+    assert crt([656, 350], [811, 133], symmetric=True) == (-56917, 114800)
+
+
+def test_modular():
+    assert solve_congruence(*list(zip([3, 4, 2], [12, 35, 17]))) == (1719, 7140)
+    assert solve_congruence(*list(zip([3, 4, 2], [12, 6, 17]))) is None
+    assert solve_congruence(*list(zip([3, 4, 2], [13, 7, 17]))) == (172, 1547)
+    assert solve_congruence(*list(zip([-10, -3, -15], [13, 7, 17]))) == (172, 1547)
+    assert solve_congruence(*list(zip([-10, -3, 1, -15], [13, 7, 7, 17]))) is None
+    assert solve_congruence(
+        *list(zip([-10, -5, 2, -15], [13, 7, 7, 17]))) == (835, 1547)
+    assert solve_congruence(
+        *list(zip([-10, -5, 2, -15], [13, 7, 14, 17]))) == (2382, 3094)
+    assert solve_congruence(
+        *list(zip([-10, 2, 2, -15], [13, 7, 14, 17]))) == (2382, 3094)
+    assert solve_congruence(*list(zip((1, 1, 2), (3, 2, 4)))) is None
+    raises(
+        ValueError, lambda: solve_congruence(*list(zip([3, 4, 2], [12.1, 35, 17]))))

rl/Lib/site-packages/sympy/ntheory/tests/test_multinomial.py

@@ -0,0 +1,48 @@
+from sympy.ntheory.multinomial import (binomial_coefficients, binomial_coefficients_list, multinomial_coefficients)
+from sympy.ntheory.multinomial import multinomial_coefficients_iterator
+
+
+def test_binomial_coefficients_list():
+    assert binomial_coefficients_list(0) == [1]
+    assert binomial_coefficients_list(1) == [1, 1]
+    assert binomial_coefficients_list(2) == [1, 2, 1]
+    assert binomial_coefficients_list(3) == [1, 3, 3, 1]
+    assert binomial_coefficients_list(4) == [1, 4, 6, 4, 1]
+    assert binomial_coefficients_list(5) == [1, 5, 10, 10, 5, 1]
+    assert binomial_coefficients_list(6) == [1, 6, 15, 20, 15, 6, 1]
+
+
+def test_binomial_coefficients():
+    for n in range(15):
+        c = binomial_coefficients(n)
+        l = [c[k] for k in sorted(c)]
+        assert l == binomial_coefficients_list(n)
+
+
+def test_multinomial_coefficients():
+    assert multinomial_coefficients(1, 1) == {(1,): 1}
+    assert multinomial_coefficients(1, 2) == {(2,): 1}
+    assert multinomial_coefficients(1, 3) == {(3,): 1}
+    assert multinomial_coefficients(2, 0) == {(0, 0): 1}
+    assert multinomial_coefficients(2, 1) == {(0, 1): 1, (1, 0): 1}
+    assert multinomial_coefficients(2, 2) == {(2, 0): 1, (0, 2): 1, (1, 1): 2}
+    assert multinomial_coefficients(2, 3) == {(3, 0): 1, (1, 2): 3, (0, 3): 1,
+            (2, 1): 3}
+    assert multinomial_coefficients(3, 1) == {(1, 0, 0): 1, (0, 1, 0): 1,
+            (0, 0, 1): 1}
+    assert multinomial_coefficients(3, 2) == {(0, 1, 1): 2, (0, 0, 2): 1,
+            (1, 1, 0): 2, (0, 2, 0): 1, (1, 0, 1): 2, (2, 0, 0): 1}
+    mc = multinomial_coefficients(3, 3)
+    assert mc == {(2, 1, 0): 3, (0, 3, 0): 1,
+            (1, 0, 2): 3, (0, 2, 1): 3, (0, 1, 2): 3, (3, 0, 0): 1,
+            (2, 0, 1): 3, (1, 2, 0): 3, (1, 1, 1): 6, (0, 0, 3): 1}
+    assert dict(multinomial_coefficients_iterator(2, 0)) == {(0, 0): 1}
+    assert dict(
+        multinomial_coefficients_iterator(2, 1)) == {(0, 1): 1, (1, 0): 1}
+    assert dict(multinomial_coefficients_iterator(2, 2)) == \
+        {(2, 0): 1, (0, 2): 1, (1, 1): 2}
+    assert dict(multinomial_coefficients_iterator(3, 3)) == mc
+    it = multinomial_coefficients_iterator(7, 2)
+    assert [next(it) for i in range(4)] == \
+        [((2, 0, 0, 0, 0, 0, 0), 1), ((1, 1, 0, 0, 0, 0, 0), 2),
+      ((0, 2, 0, 0, 0, 0, 0), 1), ((1, 0, 1, 0, 0, 0, 0), 2)]

rl/Lib/site-packages/sympy/ntheory/tests/test_partitions.py

@@ -0,0 +1,27 @@
+from sympy.ntheory.partitions_ import npartitions, _partition_rec, _partition
+
+
+def test__partition_rec():
+    A000041 = [1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 77, 101, 135,
+               176, 231, 297, 385, 490, 627, 792, 1002, 1255, 1575]
+    for n, val in enumerate(A000041):
+        assert _partition_rec(n) == val
+
+
+def test__partition():
+    assert [_partition(k) for k in range(13)] == \
+        [1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 77]
+    assert _partition(100) == 190569292
+    assert _partition(200) == 3972999029388
+    assert _partition(1000) == 24061467864032622473692149727991
+    assert _partition(1001) == 25032297938763929621013218349796
+    assert _partition(2000) == 4720819175619413888601432406799959512200344166
+    assert _partition(10000) % 10**10 == 6916435144
+    assert _partition(100000) % 10**10 == 9421098519
+
+
+def test_deprecated_ntheory_symbolic_functions():
+    from sympy.testing.pytest import warns_deprecated_sympy
+
+    with warns_deprecated_sympy():
+        assert npartitions(0) == 1

rl/Lib/site-packages/sympy/ntheory/tests/test_primetest.py

@@ -0,0 +1,230 @@
+from math import gcd
+
+from sympy.ntheory.generate import Sieve, sieve
+from sympy.ntheory.primetest import (mr, _lucas_extrastrong_params, is_lucas_prp, is_square,
+                                     is_strong_lucas_prp, is_extra_strong_lucas_prp,
+                                     proth_test, isprime, is_euler_pseudoprime,
+                                     is_gaussian_prime, is_fermat_pseudoprime, is_euler_jacobi_pseudoprime,
+                                     MERSENNE_PRIME_EXPONENTS, _lucas_lehmer_primality_test,
+                                     is_mersenne_prime)
+
+from sympy.testing.pytest import slow, raises
+from sympy.core.numbers import I, Float
+
+
+def test_is_fermat_pseudoprime():
+    assert is_fermat_pseudoprime(5, 1)
+    assert is_fermat_pseudoprime(9, 1)
+
+
+def test_euler_pseudoprimes():
+    assert is_euler_pseudoprime(13, 1)
+    assert is_euler_pseudoprime(15, 1)
+    assert is_euler_pseudoprime(17, 6)
+    assert is_euler_pseudoprime(101, 7)
+    assert is_euler_pseudoprime(1009, 10)
+    assert is_euler_pseudoprime(11287, 41)
+
+    raises(ValueError, lambda: is_euler_pseudoprime(0, 4))
+    raises(ValueError, lambda: is_euler_pseudoprime(3, 0))
+    raises(ValueError, lambda: is_euler_pseudoprime(15, 6))
+
+    # A006970
+    euler_prp = [341, 561, 1105, 1729, 1905, 2047, 2465, 3277,
+                 4033, 4681, 5461, 6601, 8321, 8481, 10261, 10585]
+    for p in euler_prp:
+        assert is_euler_pseudoprime(p, 2)
+
+    # A048950
+    euler_prp = [121, 703, 1729, 1891, 2821, 3281, 7381, 8401, 8911, 10585,
+                 12403, 15457, 15841, 16531, 18721, 19345, 23521, 24661, 28009]
+    for p in euler_prp:
+        assert is_euler_pseudoprime(p, 3)
+
+    # A033181
+    absolute_euler_prp = [1729, 2465, 15841, 41041, 46657, 75361,
+                          162401, 172081, 399001, 449065, 488881]
+    for p in absolute_euler_prp:
+        for a in range(2, p):
+            if gcd(a, p) != 1:
+                continue
+            assert is_euler_pseudoprime(p, a)
+
+
+def test_is_euler_jacobi_pseudoprime():
+    assert is_euler_jacobi_pseudoprime(11, 1)
+    assert is_euler_jacobi_pseudoprime(15, 1)
+
+
+def test_lucas_extrastrong_params():
+    assert _lucas_extrastrong_params(3) == (5, 3, 1)
+    assert _lucas_extrastrong_params(5) == (12, 4, 1)
+    assert _lucas_extrastrong_params(7) == (5, 3, 1)
+    assert _lucas_extrastrong_params(9) == (0, 0, 0)
+    assert _lucas_extrastrong_params(11) == (21, 5, 1)
+    assert _lucas_extrastrong_params(59) == (32, 6, 1)
+    assert _lucas_extrastrong_params(479) == (117, 11, 1)
+
+
+def test_is_extra_strong_lucas_prp():
+    assert is_extra_strong_lucas_prp(4) == False
+    assert is_extra_strong_lucas_prp(989) == True
+    assert is_extra_strong_lucas_prp(10877) == True
+    assert is_extra_strong_lucas_prp(9) == False
+    assert is_extra_strong_lucas_prp(16) == False
+    assert is_extra_strong_lucas_prp(169) == False
+
+@slow
+def test_prps():
+    oddcomposites = [n for n in range(1, 10**5) if
+        n % 2 and not isprime(n)]
+    # A checksum would be better.
+    assert sum(oddcomposites) == 2045603465
+    assert [n for n in oddcomposites if mr(n, [2])] == [
+        2047, 3277, 4033, 4681, 8321, 15841, 29341, 42799, 49141,
+        52633, 65281, 74665, 80581, 85489, 88357, 90751]
+    assert [n for n in oddcomposites if mr(n, [3])] == [
+        121, 703, 1891, 3281, 8401, 8911, 10585, 12403, 16531,
+        18721, 19345, 23521, 31621, 44287, 47197, 55969, 63139,
+        74593, 79003, 82513, 87913, 88573, 97567]
+    assert [n for n in oddcomposites if mr(n, [325])] == [
+        9, 25, 27, 49, 65, 81, 325, 341, 343, 697, 1141, 2059,
+        2149, 3097, 3537, 4033, 4681, 4941, 5833, 6517, 7987, 8911,
+        12403, 12913, 15043, 16021, 20017, 22261, 23221, 24649,
+        24929, 31841, 35371, 38503, 43213, 44173, 47197, 50041,
+        55909, 56033, 58969, 59089, 61337, 65441, 68823, 72641,
+        76793, 78409, 85879]
+    assert not any(mr(n, [9345883071009581737]) for n in oddcomposites)
+    assert [n for n in oddcomposites if is_lucas_prp(n)] == [
+        323, 377, 1159, 1829, 3827, 5459, 5777, 9071, 9179, 10877,
+        11419, 11663, 13919, 14839, 16109, 16211, 18407, 18971,
+        19043, 22499, 23407, 24569, 25199, 25877, 26069, 27323,
+        32759, 34943, 35207, 39059, 39203, 39689, 40309, 44099,
+        46979, 47879, 50183, 51983, 53663, 56279, 58519, 60377,
+        63881, 69509, 72389, 73919, 75077, 77219, 79547, 79799,
+        82983, 84419, 86063, 90287, 94667, 97019, 97439]
+    assert [n for n in oddcomposites if is_strong_lucas_prp(n)] == [
+        5459, 5777, 10877, 16109, 18971, 22499, 24569, 25199, 40309,
+        58519, 75077, 97439]
+    assert [n for n in oddcomposites if is_extra_strong_lucas_prp(n)
+            ] == [
+        989, 3239, 5777, 10877, 27971, 29681, 30739, 31631, 39059,
+        72389, 73919, 75077]
+
+
+def test_proth_test():
+    # Proth number
+    A080075 = [3, 5, 9, 13, 17, 25, 33, 41, 49, 57, 65,
+               81, 97, 113, 129, 145, 161, 177, 193]
+    # Proth prime
+    A080076 = [3, 5, 13, 17, 41, 97, 113, 193]
+
+    for n in range(200):
+        if n in A080075:
+            assert proth_test(n) == (n in A080076)
+        else:
+            raises(ValueError, lambda: proth_test(n))
+
+
+def test_lucas_lehmer_primality_test():
+    for p in sieve.primerange(3, 100):
+        assert _lucas_lehmer_primality_test(p) == (p in MERSENNE_PRIME_EXPONENTS)
+
+
+def test_is_mersenne_prime():
+    assert is_mersenne_prime(-3) is False
+    assert is_mersenne_prime(3) is True
+    assert is_mersenne_prime(10) is False
+    assert is_mersenne_prime(127) is True
+    assert is_mersenne_prime(511) is False
+    assert is_mersenne_prime(131071) is True
+    assert is_mersenne_prime(2147483647) is True
+
+
+def test_isprime():
+    s = Sieve()
+    s.extend(100000)
+    ps = set(s.primerange(2, 100001))
+    for n in range(100001):
+        # if (n in ps) != isprime(n): print n
+        assert (n in ps) == isprime(n)
+    assert isprime(179424673)
+    assert isprime(20678048681)
+    assert isprime(1968188556461)
+    assert isprime(2614941710599)
+    assert isprime(65635624165761929287)
+    assert isprime(1162566711635022452267983)
+    assert isprime(77123077103005189615466924501)
+    assert isprime(3991617775553178702574451996736229)
+    assert isprime(273952953553395851092382714516720001799)
+    assert isprime(int('''
+531137992816767098689588206552468627329593117727031923199444138200403\
+559860852242739162502265229285668889329486246501015346579337652707239\
+409519978766587351943831270835393219031728127'''))
+
+    # Some Mersenne primes
+    assert isprime(2**61 - 1)
+    assert isprime(2**89 - 1)
+    assert isprime(2**607 - 1)
+    # (but not all Mersenne's are primes
+    assert not isprime(2**601 - 1)
+
+    # pseudoprimes
+    #-------------
+    # to some small bases
+    assert not isprime(2152302898747)
+    assert not isprime(3474749660383)
+    assert not isprime(341550071728321)
+    assert not isprime(3825123056546413051)
+    # passes the base set [2, 3, 7, 61, 24251]
+    assert not isprime(9188353522314541)
+    # large examples
+    assert not isprime(877777777777777777777777)
+    # conjectured psi_12 given at http://mathworld.wolfram.com/StrongPseudoprime.html
+    assert not isprime(318665857834031151167461)
+    # conjectured psi_17 given at http://mathworld.wolfram.com/StrongPseudoprime.html
+    assert not isprime(564132928021909221014087501701)
+    # Arnault's 1993 number; a factor of it is
+    # 400958216639499605418306452084546853005188166041132508774506\
+    # 204738003217070119624271622319159721973358216316508535816696\
+    # 9145233813917169287527980445796800452592031836601
+    assert not isprime(int('''
+803837457453639491257079614341942108138837688287558145837488917522297\
+427376533365218650233616396004545791504202360320876656996676098728404\
+396540823292873879185086916685732826776177102938969773947016708230428\
+687109997439976544144845341155872450633409279022275296229414984230688\
+1685404326457534018329786111298960644845216191652872597534901'''))
+    # Arnault's 1995 number; can be factored as
+    # p1*(313*(p1 - 1) + 1)*(353*(p1 - 1) + 1) where p1 is
+    # 296744956686855105501541746429053327307719917998530433509950\
+    # 755312768387531717701995942385964281211880336647542183455624\
+    # 93168782883
+    assert not isprime(int('''
+288714823805077121267142959713039399197760945927972270092651602419743\
+230379915273311632898314463922594197780311092934965557841894944174093\
+380561511397999942154241693397290542371100275104208013496673175515285\
+922696291677532547504444585610194940420003990443211677661994962953925\
+045269871932907037356403227370127845389912612030924484149472897688540\
+6024976768122077071687938121709811322297802059565867'''))
+    sieve.extend(3000)
+    assert isprime(2819)
+    assert not isprime(2931)
+    raises(ValueError, lambda: isprime(2.0))
+    raises(ValueError, lambda: isprime(Float(2)))
+
+
+def test_is_square():
+    assert [i for i in range(25) if is_square(i)] == [0, 1, 4, 9, 16]
+
+    # issue #17044
+    assert not is_square(60 ** 3)
+    assert not is_square(60 ** 5)
+    assert not is_square(84 ** 7)
+    assert not is_square(105 ** 9)
+    assert not is_square(120 ** 3)
+
+def test_is_gaussianprime():
+    assert is_gaussian_prime(7*I)
+    assert is_gaussian_prime(7)
+    assert is_gaussian_prime(2 + 3*I)
+    assert not is_gaussian_prime(2 + 2*I)

rl/Lib/site-packages/sympy/ntheory/tests/test_qs.py

@@ -0,0 +1,124 @@
+from __future__ import annotations
+
+from sympy.ntheory import qs
+from sympy.ntheory.qs import SievePolynomial, _generate_factor_base, \
+    _initialize_first_polynomial, _initialize_ith_poly, \
+    _gen_sieve_array, _check_smoothness, _trial_division_stage, _gauss_mod_2, \
+    _build_matrix, _find_factor
+from sympy.testing.pytest import slow
+
+
+@slow
+def test_qs_1():
+    assert qs(10009202107, 100, 10000) == {100043, 100049}
+    assert qs(211107295182713951054568361, 1000, 10000) == \
+        {13791315212531, 15307263442931}
+    assert qs(980835832582657*990377764891511, 3000, 50000) == \
+        {980835832582657, 990377764891511}
+    assert qs(18640889198609*20991129234731, 1000, 50000) == \
+        {18640889198609, 20991129234731}
+
+
+def test_qs_2() -> None:
+    n = 10009202107
+    M = 50
+    # a = 10, b = 15, modified_coeff = [a**2, 2*a*b, b**2 - N]
+    sieve_poly = SievePolynomial([100,  1600, -10009195707], 10, 80)
+    assert sieve_poly.eval(10) == -10009169707
+    assert sieve_poly.eval(5) == -10009185207
+
+    idx_1000, idx_5000, factor_base = _generate_factor_base(2000, n)
+    assert idx_1000 == 82
+    assert [factor_base[i].prime for i in range(15)] == \
+        [2, 3, 7, 11, 17, 19, 29, 31, 43, 59, 61, 67, 71, 73, 79]
+    assert [factor_base[i].tmem_p for i in range(15)] == \
+        [1, 1, 3, 5, 3, 6, 6, 14, 1, 16, 24, 22, 18, 22, 15]
+    assert [factor_base[i].log_p for i in range(5)] == \
+        [710, 1125, 1993, 2455, 2901]
+
+    g, B = _initialize_first_polynomial(
+        n, M, factor_base, idx_1000, idx_5000, seed=0)
+    assert g.a == 1133107
+    assert g.b == 682543
+    assert B == [272889, 409654]
+    assert [factor_base[i].soln1 for i in range(15)] == \
+        [0, 0, 3, 7, 13, 0, 8, 19, 9, 43, 27, 25, 63, 29, 19]
+    assert [factor_base[i].soln2 for i in range(15)] == \
+        [0, 1, 1, 3, 12, 16, 15, 6, 15, 1, 56, 55, 61, 58, 16]
+    assert [factor_base[i].a_inv for i in range(15)] == \
+        [1, 1, 5, 7, 3, 5, 26, 6, 40, 5, 21, 45, 4, 1, 8]
+    assert [factor_base[i].b_ainv for i in range(5)] == \
+        [[0, 0], [0, 2], [3, 0], [3, 9], [13, 13]]
+
+    g_1 = _initialize_ith_poly(n, factor_base, 1, g, B)
+    assert g_1.a == 1133107
+    assert g_1.b == 136765
+
+    sieve_array = _gen_sieve_array(M, factor_base)
+    assert sieve_array[0:5] == [8424, 13603, 1835, 5335, 710]
+
+    assert _check_smoothness(9645, factor_base) == (5, False)
+    assert _check_smoothness(210313, factor_base)[0][0:15] == \
+        [0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 1]
+    assert _check_smoothness(210313, factor_base)[1]
+
+    partial_relations: dict[int, tuple[int, int]] = {}
+    smooth_relation, partial_relation = _trial_division_stage(
+        n, M, factor_base, sieve_array, sieve_poly, partial_relations,
+        ERROR_TERM=25*2**10)
+
+    assert partial_relations == {
+        8699: (440, -10009008507),
+        166741: (490, -10008962007),
+        131449: (530, -10008921207),
+        6653: (550, -10008899607)
+    }
+    assert [smooth_relation[i][0] for i in range(5)] == [
+        -250, -670615476700, -45211565844500, -231723037747200, -1811665537200]
+    assert [smooth_relation[i][1] for i in range(5)] == [
+        -10009139607, 1133094251961, 5302606761, 53804049849, 1950723889]
+    assert smooth_relation[0][2][0:15] == [
+        1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
+
+    assert _gauss_mod_2(
+        [[0, 0, 1], [1, 0, 1], [0, 1, 0], [0, 1, 1], [0, 1, 1]]
+    ) == (
+        [[[0, 1, 1], 3], [[0, 1, 1], 4]],
+        [True, True, True, False, False],
+        [[0, 0, 1], [1, 0, 0], [0, 1, 0], [0, 1, 1], [0, 1, 1]]
+    )
+
+
+def test_qs_3():
+    N = 1817
+    smooth_relations = [
+        (2455024, 637, [0, 0, 0, 1]),
+        (-27993000, 81536, [0, 1, 0, 1]),
+        (11461840, 12544, [0, 0, 0, 0]),
+        (149, 20384, [0, 1, 0, 1]),
+        (-31138074, 19208, [0, 1, 0, 0])
+    ]
+
+    matrix = _build_matrix(smooth_relations)
+    assert matrix == [
+        [0, 0, 0, 1],
+        [0, 1, 0, 1],
+        [0, 0, 0, 0],
+        [0, 1, 0, 1],
+        [0, 1, 0, 0]
+    ]
+
+    dependent_row, mark, gauss_matrix = _gauss_mod_2(matrix)
+    assert dependent_row == [[[0, 0, 0, 0], 2], [[0, 1, 0, 0], 3]]
+    assert mark == [True, True, False, False, True]
+    assert gauss_matrix == [
+        [0, 0, 0, 1],
+        [0, 1, 0, 0],
+        [0, 0, 0, 0],
+        [0, 1, 0, 0],
+        [0, 1, 0, 1]
+    ]
+
+    factor = _find_factor(
+        dependent_row, mark, gauss_matrix, 0, smooth_relations, N)
+    assert factor == 23

rl/Lib/site-packages/sympy/ntheory/tests/test_residue.py

@@ -0,0 +1,345 @@
+from collections import defaultdict
+from sympy.core.containers import Tuple
+from sympy.core.singleton import S
+from sympy.core.symbol import (Dummy, Symbol)
+from sympy.functions.combinatorial.numbers import totient
+from sympy.ntheory import n_order, is_primitive_root, is_quad_residue, \
+    legendre_symbol, jacobi_symbol, primerange, sqrt_mod, \
+    primitive_root, quadratic_residues, is_nthpow_residue, nthroot_mod, \
+    sqrt_mod_iter, mobius, discrete_log, quadratic_congruence, \
+    polynomial_congruence, sieve
+from sympy.ntheory.residue_ntheory import _primitive_root_prime_iter, \
+    _primitive_root_prime_power_iter, _primitive_root_prime_power2_iter, \
+    _nthroot_mod_prime_power, _discrete_log_trial_mul, _discrete_log_shanks_steps, \
+    _discrete_log_pollard_rho, _discrete_log_index_calculus, _discrete_log_pohlig_hellman, \
+    _binomial_mod_prime_power, binomial_mod
+from sympy.polys.domains import ZZ
+from sympy.testing.pytest import raises
+from sympy.core.random import randint, choice
+
+
+def test_residue():
+    assert n_order(2, 13) == 12
+    assert [n_order(a, 7) for a in range(1, 7)] == \
+           [1, 3, 6, 3, 6, 2]
+    assert n_order(5, 17) == 16
+    assert n_order(17, 11) == n_order(6, 11)
+    assert n_order(101, 119) == 6
+    assert n_order(11, (10**50 + 151)**2) == 10000000000000000000000000000000000000000000000030100000000000000000000000000000000000000000000022650
+    raises(ValueError, lambda: n_order(6, 9))
+
+    assert is_primitive_root(2, 7) is False
+    assert is_primitive_root(3, 8) is False
+    assert is_primitive_root(11, 14) is False
+    assert is_primitive_root(12, 17) == is_primitive_root(29, 17)
+    raises(ValueError, lambda: is_primitive_root(3, 6))
+
+    for p in primerange(3, 100):
+        li = list(_primitive_root_prime_iter(p))
+        assert li[0] == min(li)
+        for g in li:
+            assert n_order(g, p) == p - 1
+        assert len(li) == totient(totient(p))
+        for e in range(1, 4):
+            li_power = list(_primitive_root_prime_power_iter(p, e))
+            li_power2 = list(_primitive_root_prime_power2_iter(p, e))
+            assert len(li_power) == len(li_power2) == totient(totient(p**e))
+    assert primitive_root(97) == 5
+    assert n_order(primitive_root(97, False), 97) == totient(97)
+    assert primitive_root(97**2) == 5
+    assert n_order(primitive_root(97**2, False), 97**2) == totient(97**2)
+    assert primitive_root(40487) == 5
+    assert n_order(primitive_root(40487, False), 40487) == totient(40487)
+    # note that primitive_root(40487) + 40487 = 40492 is a primitive root
+    # of 40487**2, but it is not the smallest
+    assert primitive_root(40487**2) == 10
+    assert n_order(primitive_root(40487**2, False), 40487**2) == totient(40487**2)
+    assert primitive_root(82) == 7
+    assert n_order(primitive_root(82, False), 82) == totient(82)
+    p = 10**50 + 151
+    assert primitive_root(p) == 11
+    assert n_order(primitive_root(p, False), p) == totient(p)
+    assert primitive_root(2*p) == 11
+    assert n_order(primitive_root(2*p, False), 2*p) == totient(2*p)
+    assert primitive_root(p**2) == 11
+    assert n_order(primitive_root(p**2, False), p**2) == totient(p**2)
+    assert primitive_root(4 * 11) is None and primitive_root(4 * 11, False) is None
+    assert primitive_root(15) is None and primitive_root(15, False) is None
+    raises(ValueError, lambda: primitive_root(-3))
+
+    assert is_quad_residue(3, 7) is False
+    assert is_quad_residue(10, 13) is True
+    assert is_quad_residue(12364, 139) == is_quad_residue(12364 % 139, 139)
+    assert is_quad_residue(207, 251) is True
+    assert is_quad_residue(0, 1) is True
+    assert is_quad_residue(1, 1) is True
+    assert is_quad_residue(0, 2) == is_quad_residue(1, 2) is True
+    assert is_quad_residue(1, 4) is True
+    assert is_quad_residue(2, 27) is False
+    assert is_quad_residue(13122380800, 13604889600) is True
+    assert [j for j in range(14) if is_quad_residue(j, 14)] == \
+           [0, 1, 2, 4, 7, 8, 9, 11]
+    raises(ValueError, lambda: is_quad_residue(1.1, 2))
+    raises(ValueError, lambda: is_quad_residue(2, 0))
+
+    assert quadratic_residues(S.One) == [0]
+    assert quadratic_residues(1) == [0]
+    assert quadratic_residues(12) == [0, 1, 4, 9]
+    assert quadratic_residues(13) == [0, 1, 3, 4, 9, 10, 12]
+    assert [len(quadratic_residues(i)) for i in range(1, 20)] == \
+      [1, 2, 2, 2, 3, 4, 4, 3, 4, 6, 6, 4, 7, 8, 6, 4, 9, 8, 10]
+
+    assert list(sqrt_mod_iter(6, 2)) == [0]
+    assert sqrt_mod(3, 13) == 4
+    assert sqrt_mod(3, -13) == 4
+    assert sqrt_mod(6, 23) == 11
+    assert sqrt_mod(345, 690) == 345
+    assert sqrt_mod(67, 101) == None
+    assert sqrt_mod(1020, 104729) == None
+
+    for p in range(3, 100):
+        d = defaultdict(list)
+        for i in range(p):
+            d[pow(i, 2, p)].append(i)
+        for i in range(1, p):
+            it = sqrt_mod_iter(i, p)
+            v = sqrt_mod(i, p, True)
+            if v:
+                v = sorted(v)
+                assert d[i] == v
+            else:
+                assert not d[i]
+
+    assert sqrt_mod(9, 27, True) == [3, 6, 12, 15, 21, 24]
+    assert sqrt_mod(9, 81, True) == [3, 24, 30, 51, 57, 78]
+    assert sqrt_mod(9, 3**5, True) == [3, 78, 84, 159, 165, 240]
+    assert sqrt_mod(81, 3**4, True) == [0, 9, 18, 27, 36, 45, 54, 63, 72]
+    assert sqrt_mod(81, 3**5, True) == [9, 18, 36, 45, 63, 72, 90, 99, 117,\
+            126, 144, 153, 171, 180, 198, 207, 225, 234]
+    assert sqrt_mod(81, 3**6, True) == [9, 72, 90, 153, 171, 234, 252, 315,\
+            333, 396, 414, 477, 495, 558, 576, 639, 657, 720]
+    assert sqrt_mod(81, 3**7, True) == [9, 234, 252, 477, 495, 720, 738, 963,\
+            981, 1206, 1224, 1449, 1467, 1692, 1710, 1935, 1953, 2178]
+
+    for a, p in [(26214400, 32768000000), (26214400, 16384000000),
+        (262144, 1048576), (87169610025, 163443018796875),
+        (22315420166400, 167365651248000000)]:
+        assert pow(sqrt_mod(a, p), 2, p) == a
+
+    n = 70
+    a, p = 5**2*3**n*2**n, 5**6*3**(n+1)*2**(n+2)
+    it = sqrt_mod_iter(a, p)
+    for i in range(10):
+        assert pow(next(it), 2, p) == a
+    a, p = 5**2*3**n*2**n, 5**6*3**(n+1)*2**(n+3)
+    it = sqrt_mod_iter(a, p)
+    for i in range(2):
+        assert pow(next(it), 2, p) == a
+    n = 100
+    a, p = 5**2*3**n*2**n, 5**6*3**(n+1)*2**(n+1)
+    it = sqrt_mod_iter(a, p)
+    for i in range(2):
+        assert pow(next(it), 2, p) == a
+
+    assert type(next(sqrt_mod_iter(9, 27))) is int
+    assert type(next(sqrt_mod_iter(9, 27, ZZ))) is type(ZZ(1))
+    assert type(next(sqrt_mod_iter(1, 7, ZZ))) is type(ZZ(1))
+
+    assert is_nthpow_residue(2, 1, 5)
+
+    #issue 10816
+    assert is_nthpow_residue(1, 0, 1) is False
+    assert is_nthpow_residue(1, 0, 2) is True
+    assert is_nthpow_residue(3, 0, 2) is True
+    assert is_nthpow_residue(0, 1, 8) is True
+    assert is_nthpow_residue(2, 3, 2) is True
+    assert is_nthpow_residue(2, 3, 9) is False
+    assert is_nthpow_residue(3, 5, 30) is True
+    assert is_nthpow_residue(21, 11, 20) is True
+    assert is_nthpow_residue(7, 10, 20) is False
+    assert is_nthpow_residue(5, 10, 20) is True
+    assert is_nthpow_residue(3, 10, 48) is False
+    assert is_nthpow_residue(1, 10, 40) is True
+    assert is_nthpow_residue(3, 10, 24) is False
+    assert is_nthpow_residue(1, 10, 24) is True
+    assert is_nthpow_residue(3, 10, 24) is False
+    assert is_nthpow_residue(2, 10, 48) is False
+    assert is_nthpow_residue(81, 3, 972) is False
+    assert is_nthpow_residue(243, 5, 5103) is True
+    assert is_nthpow_residue(243, 3, 1240029) is False
+    assert is_nthpow_residue(36010, 8, 87382) is True
+    assert is_nthpow_residue(28552, 6, 2218) is True
+    assert is_nthpow_residue(92712, 9, 50026) is True
+    x = {pow(i, 56, 1024) for i in range(1024)}
+    assert {a for a in range(1024) if is_nthpow_residue(a, 56, 1024)} == x
+    x = { pow(i, 256, 2048) for i in range(2048)}
+    assert {a for a in range(2048) if is_nthpow_residue(a, 256, 2048)} == x
+    x = { pow(i, 11, 324000) for i in range(1000)}
+    assert [ is_nthpow_residue(a, 11, 324000) for a in x]
+    x = { pow(i, 17, 22217575536) for i in range(1000)}
+    assert [ is_nthpow_residue(a, 17, 22217575536) for a in x]
+    assert is_nthpow_residue(676, 3, 5364)
+    assert is_nthpow_residue(9, 12, 36)
+    assert is_nthpow_residue(32, 10, 41)
+    assert is_nthpow_residue(4, 2, 64)
+    assert is_nthpow_residue(31, 4, 41)
+    assert not is_nthpow_residue(2, 2, 5)
+    assert is_nthpow_residue(8547, 12, 10007)
+    assert is_nthpow_residue(Dummy(even=True) + 3, 3, 2) == True
+    # _nthroot_mod_prime_power
+    for p in primerange(2, 10):
+        for a in range(3):
+            for n in range(3, 5):
+                ans = _nthroot_mod_prime_power(a, n, p, 1)
+                assert isinstance(ans, list)
+                if len(ans) == 0:
+                    for b in range(p):
+                        assert pow(b, n, p) != a % p
+                    for k in range(2, 10):
+                        assert _nthroot_mod_prime_power(a, n, p, k) == []
+                else:
+                    for b in range(p):
+                        pred = pow(b, n, p) == a % p
+                        assert not(pred ^ (b in ans))
+                    for k in range(2, 10):
+                        ans = _nthroot_mod_prime_power(a, n, p, k)
+                        if not ans:
+                            break
+                        for b in ans:
+                            assert pow(b, n , p**k) == a
+
+    assert nthroot_mod(Dummy(odd=True), 3, 2) == 1
+    assert nthroot_mod(29, 31, 74) == 45
+    assert nthroot_mod(1801, 11, 2663) == 44
+    for a, q, p in [(51922, 2, 203017), (43, 3, 109), (1801, 11, 2663),
+          (26118163, 1303, 33333347), (1499, 7, 2663), (595, 6, 2663),
+          (1714, 12, 2663), (28477, 9, 33343)]:
+        r = nthroot_mod(a, q, p)
+        assert pow(r, q, p) == a
+    assert nthroot_mod(11, 3, 109) is None
+    assert nthroot_mod(16, 5, 36, True) == [4, 22]
+    assert nthroot_mod(9, 16, 36, True) == [3, 9, 15, 21, 27, 33]
+    assert nthroot_mod(4, 3, 3249000) is None
+    assert nthroot_mod(36010, 8, 87382, True) == [40208, 47174]
+    assert nthroot_mod(0, 12, 37, True) == [0]
+    assert nthroot_mod(0, 7, 100, True) == [0, 10, 20, 30, 40, 50, 60, 70, 80, 90]
+    assert nthroot_mod(4, 4, 27, True) == [5, 22]
+    assert nthroot_mod(4, 4, 121, True) == [19, 102]
+    assert nthroot_mod(2, 3, 7, True) == []
+    for p in range(1, 20):
+        for a in range(p):
+            for n in range(1, p):
+                ans = nthroot_mod(a, n, p, True)
+                assert isinstance(ans, list)
+                for b in range(p):
+                    pred = pow(b, n, p) == a
+                    assert not(pred ^ (b in ans))
+                ans2 = nthroot_mod(a, n, p, False)
+                if ans2 is None:
+                    assert ans == []
+                else:
+                    assert ans2 in ans
+
+    x = Symbol('x', positive=True)
+    i = Symbol('i', integer=True)
+    assert _discrete_log_trial_mul(587, 2**7, 2) == 7
+    assert _discrete_log_trial_mul(941, 7**18, 7) == 18
+    assert _discrete_log_trial_mul(389, 3**81, 3) == 81
+    assert _discrete_log_trial_mul(191, 19**123, 19) == 123
+    assert _discrete_log_shanks_steps(442879, 7**2, 7) == 2
+    assert _discrete_log_shanks_steps(874323, 5**19, 5) == 19
+    assert _discrete_log_shanks_steps(6876342, 7**71, 7) == 71
+    assert _discrete_log_shanks_steps(2456747, 3**321, 3) == 321
+    assert _discrete_log_pollard_rho(6013199, 2**6, 2, rseed=0) == 6
+    assert _discrete_log_pollard_rho(6138719, 2**19, 2, rseed=0) == 19
+    assert _discrete_log_pollard_rho(36721943, 2**40, 2, rseed=0) == 40
+    assert _discrete_log_pollard_rho(24567899, 3**333, 3, rseed=0) == 333
+    raises(ValueError, lambda: _discrete_log_pollard_rho(11, 7, 31, rseed=0))
+    raises(ValueError, lambda: _discrete_log_pollard_rho(227, 3**7, 5, rseed=0))
+    assert _discrete_log_index_calculus(983, 948, 2, 491) == 183
+    assert _discrete_log_index_calculus(633383, 21794, 2, 316691) == 68048
+    assert _discrete_log_index_calculus(941762639, 68822582, 2, 470881319) == 338029275
+    assert _discrete_log_index_calculus(999231337607, 888188918786, 2, 499615668803) == 142811376514
+    assert _discrete_log_index_calculus(47747730623, 19410045286, 43425105668, 645239603) == 590504662
+    assert _discrete_log_pohlig_hellman(98376431, 11**9, 11) == 9
+    assert _discrete_log_pohlig_hellman(78723213, 11**31, 11) == 31
+    assert _discrete_log_pohlig_hellman(32942478, 11**98, 11) == 98
+    assert _discrete_log_pohlig_hellman(14789363, 11**444, 11) == 444
+    assert discrete_log(587, 2**9, 2) == 9
+    assert discrete_log(2456747, 3**51, 3) == 51
+    assert discrete_log(32942478, 11**127, 11) == 127
+    assert discrete_log(432751500361, 7**324, 7) == 324
+    assert discrete_log(265390227570863,184500076053622, 2) == 17835221372061
+    assert discrete_log(22708823198678103974314518195029102158525052496759285596453269189798311427475159776411276642277139650833937,
+                        17463946429475485293747680247507700244427944625055089103624311227422110546803452417458985046168310373075327,
+                        123456) == 2068031853682195777930683306640554533145512201725884603914601918777510185469769997054750835368413389728895
+    args = 5779, 3528, 6215
+    assert discrete_log(*args) == 687
+    assert discrete_log(*Tuple(*args)) == 687
+    assert quadratic_congruence(400, 85, 125, 1600) == [295, 615, 935, 1255, 1575]
+    assert quadratic_congruence(3, 6, 5, 25) == [3, 20]
+    assert quadratic_congruence(120, 80, 175, 500) == []
+    assert quadratic_congruence(15, 14, 7, 2) == [1]
+    assert quadratic_congruence(8, 15, 7, 29) == [10, 28]
+    assert quadratic_congruence(160, 200, 300, 461) == [144, 431]
+    assert quadratic_congruence(100000, 123456, 7415263, 48112959837082048697) == [30417843635344493501, 36001135160550533083]
+    assert quadratic_congruence(65, 121, 72, 277) == [249, 252]
+    assert quadratic_congruence(5, 10, 14, 2) == [0]
+    assert quadratic_congruence(10, 17, 19, 2) == [1]
+    assert quadratic_congruence(10, 14, 20, 2) == [0, 1]
+    assert polynomial_congruence(6*x**5 + 10*x**4 + 5*x**3 + x**2 + x + 1,
+        972000) == [220999, 242999, 463999, 485999, 706999, 728999, 949999, 971999]
+
+    assert polynomial_congruence(x**3 - 10*x**2 + 12*x - 82, 33075) == [30287]
+    assert polynomial_congruence(x**2 + x + 47, 2401) == [785, 1615]
+    assert polynomial_congruence(10*x**2 + 14*x + 20, 2) == [0, 1]
+    assert polynomial_congruence(x**3 + 3, 16) == [5]
+    assert polynomial_congruence(65*x**2 + 121*x + 72, 277) == [249, 252]
+    assert polynomial_congruence(x**4 - 4, 27) == [5, 22]
+    assert polynomial_congruence(35*x**3 - 6*x**2 - 567*x + 2308, 148225) == [86957,
+        111157, 122531, 146731]
+    assert polynomial_congruence(x**16 - 9, 36) == [3, 9, 15, 21, 27, 33]
+    assert polynomial_congruence(x**6 - 2*x**5 - 35, 6125) == [3257]
+    raises(ValueError, lambda: polynomial_congruence(x**x, 6125))
+    raises(ValueError, lambda: polynomial_congruence(x**i, 6125))
+    raises(ValueError, lambda: polynomial_congruence(0.1*x**2 + 6, 100))
+
+    assert binomial_mod(-1, 1, 10) == 0
+    assert binomial_mod(1, -1, 10) == 0
+    raises(ValueError, lambda: binomial_mod(2, 1, -1))
+    assert binomial_mod(51, 10, 10) == 0
+    assert binomial_mod(10**3, 500, 3**6) == 567
+    assert binomial_mod(10**18 - 1, 123456789, 4) == 0
+    assert binomial_mod(10**18, 10**12, (10**5 + 3)**2) == 3744312326
+
+
+def test_binomial_p_pow():
+    n, binomials, binomial = 1000, [1], 1
+    for i in range(1, n + 1):
+        binomial *= n - i + 1
+        binomial //= i
+        binomials.append(binomial)
+
+    # Test powers of two, which the algorithm treats slightly differently
+    trials_2 = 100
+    for _ in range(trials_2):
+        m, power = randint(0, n), randint(1, 20)
+        assert _binomial_mod_prime_power(n, m, 2, power) == binomials[m] % 2**power
+
+    # Test against other prime powers
+    primes = list(sieve.primerange(2*n))
+    trials = 1000
+    for _ in range(trials):
+        m, prime, power = randint(0, n), choice(primes), randint(1, 10)
+        assert _binomial_mod_prime_power(n, m, prime, power) == binomials[m] % prime**power
+
+
+def test_deprecated_ntheory_symbolic_functions():
+    from sympy.testing.pytest import warns_deprecated_sympy
+
+    with warns_deprecated_sympy():
+        assert mobius(3) == -1
+    with warns_deprecated_sympy():
+        assert legendre_symbol(2, 3) == -1
+    with warns_deprecated_sympy():
+        assert jacobi_symbol(2, 3) == -1