def test_diophantine_permute_sign():
from sympy.abc import a, b, c, d, e
eq = a**4 + b**4 - (2**4 + 3**4)
base_sol = set([(2, 3)])
assert diophantine(eq) == base_sol
complete_soln = set(signed_permutations(base_sol.pop()))
assert diophantine(eq, permute=True) == complete_soln
eq = a**2 + b**2 + c**2 + d**2 + e**2 - 234
assert len(diophantine(eq)) == 35
assert len(diophantine(eq, permute=True)) == 62000
soln = set([(-1, -1), (-1, 2), (1, -2), (1, 1)])
assert diophantine(10*x**2 + 12*x*y + 12*y**2 - 34, permute=True) == soln
def test_diophantine_permute_sign():
from sympy.abc import a, b, c, d, e
eq = a**4 + b**4 - (2**4 + 3**4)
base_sol = set([(2, 3)])
assert diophantine(eq) == base_sol
complete_soln = set(signed_permutations(base_sol.pop()))
assert diophantine(eq, permute=True) == complete_soln
eq = a**2 + b**2 + c**2 + d**2 + e**2 - 234
assert len(diophantine(eq)) == 35
assert len(diophantine(eq, permute=True)) == 62000
soln = set([(-1, -1), (-1, 2), (1, -2), (1, 1)])
assert diophantine(10*x**2 + 12*x*y + 12*y**2 - 34, permute=True) == soln
def _build(self) -> None:
"""Build Cayley graph and store it in self.G.
"""
self._G = nx.MultiDiGraph()
# Compute solutions to the four squares decomposition problem and
# retain only the ones eligible for the LPS construction.
four_squares = set(
[
s for x in power_representation(self.p, 2, 4, zeros=True)
for s in signed_permutations(x) if self.eligible_solution(s)
]
)
i = self.find_square_root(-1)
# One extra vertex for the infinity point
for x in range(self.q+1):
self._G.add_node(x)
for x in range(self.q):
for (a, b, c, d) in four_squares:
num = ((a + i*b) * x + c + i*d) % self.q
den = ((i*d - c) * x + a - i*b) % self.q
if den == 0:
self._G.add_edge(x, self.infinity_point)
else:
self._G.add_edge(x, num * self.modular_inv(den))
# Finally add the links from the infinity point
for (a, b, c, d) in four_squares:
num = (a + i*b) % self.q
den = (i*d - c) % self.q
if den == 0:
self._G.add_edge(
self.infinity_point, self.infinity_point
)
else:
self._G.add_edge(
self.infinity_point, num * self.modular_inv(den)
)
if self.remove_parallel_edges:
self._G = nx.Graph(self._G)
if self.remove_self_edges:
self._G.remove_edges_from(self._G.selfloop_edges())
else:
self._G = nx.MultiDiGraph(self._G)
def test_diophantine():
assert check_solutions((x - y) * (y - z) * (z - x))
assert check_solutions((x - y) * (x ** 2 + y ** 2 - z ** 2))
assert check_solutions((x - 3 * y + 7 * z) * (x ** 2 + y ** 2 - z ** 2))
assert check_solutions((x ** 2 - 3 * y ** 2 - 1))
assert check_solutions(y ** 2 + 7 * x * y)
assert check_solutions(x ** 2 - 3 * x * y + y ** 2)
assert check_solutions(z * (x ** 2 - y ** 2 - 15))
assert check_solutions(x * (2 * y - 2 * z + 5))
assert check_solutions((x ** 2 - 3 * y ** 2 - 1) * (x ** 2 - y ** 2 - 15))
assert check_solutions((x ** 2 - 3 * y ** 2 - 1) * (y - 7 * z))
assert check_solutions((x ** 2 + y ** 2 - z ** 2) * (x - 7 * y - 3 * z + 4 * w))
# Following test case caused problems in parametric representation
# But this can be solved by factoring out y.
# No need to use methods for ternary quadratic equations.
assert check_solutions(y ** 2 - 7 * x * y + 4 * y * z)
assert check_solutions(x ** 2 - 2 * x + 1)
assert diophantine(x - y) == diophantine(Eq(x, y))
# 18196
eq = x ** 4 + y ** 4 - 97
assert diophantine(eq, permute=True) == diophantine(-eq, permute=True)
assert diophantine(3 * x * pi - 2 * y * pi) == {(2 * t_0, 3 * t_0)}
eq = x ** 2 + y ** 2 + z ** 2 - 14
base_sol = {(1, 2, 3)}
assert diophantine(eq) == base_sol
complete_soln = set(signed_permutations(base_sol.pop()))
assert diophantine(eq, permute=True) == complete_soln
assert diophantine(x ** 2 + x * Rational(15, 14) - 3) == set()
# test issue 11049
eq = 92 * x ** 2 - 99 * y ** 2 - z ** 2
coeff = eq.as_coefficients_dict()
assert _diop_ternary_quadratic_normal((x, y, z), coeff) == (9, 7, 51)
assert diophantine(eq) == {
(
891 * p ** 2 + 9 * q ** 2,
-693 * p ** 2 - 102 * p * q + 7 * q ** 2,
5049 * p ** 2 - 1386 * p * q - 51 * q ** 2,
)
}
eq = 2 * x ** 2 + 2 * y ** 2 - z ** 2
coeff = eq.as_coefficients_dict()
assert _diop_ternary_quadratic_normal((x, y, z), coeff) == (1, 1, 2)
assert diophantine(eq) == {
(
2 * p ** 2 - q ** 2,
-2 * p ** 2 + 4 * p * q - q ** 2,
4 * p ** 2 - 4 * p * q + 2 * q ** 2,
)
}
eq = 411 * x ** 2 + 57 * y ** 2 - 221 * z ** 2
coeff = eq.as_coefficients_dict()
assert _diop_ternary_quadratic_normal((x, y, z), coeff) == (2021, 2645, 3066)
assert diophantine(eq) == {
(
115197 * p ** 2 - 446641 * q ** 2,
-150765 * p ** 2 + 1355172 * p * q - 584545 * q ** 2,
174762 * p ** 2 - 301530 * p * q + 677586 * q ** 2,
)
}
eq = 573 * x ** 2 + 267 * y ** 2 - 984 * z ** 2
coeff = eq.as_coefficients_dict()
assert _diop_ternary_quadratic_normal((x, y, z), coeff) == (49, 233, 127)
assert diophantine(eq) == {
(
4361 * p ** 2 - 16072 * q ** 2,
-20737 * p ** 2 + 83312 * p * q - 76424 * q ** 2,
11303 * p ** 2 - 41474 * p * q + 41656 * q ** 2,
)
}
# this produces factors during reconstruction
eq = x ** 2 + 3 * y ** 2 - 12 * z ** 2
coeff = eq.as_coefficients_dict()
assert _diop_ternary_quadratic_normal((x, y, z), coeff) == (0, 2, 1)
assert diophantine(eq) == {
(24 * p * q, 2 * p ** 2 - 24 * q ** 2, p ** 2 + 12 * q ** 2)
}
# solvers have not been written for every type
raises(NotImplementedError, lambda: diophantine(x * y ** 2 + 1))
# rational expressions
assert diophantine(1 / x) == set()
assert diophantine(1 / x + 1 / y - S.Half) == {
(6, 3),
(-2, 1),
(4, 4),
(1, -2),
(3, 6),
}
assert diophantine(x ** 2 + y ** 2 + 3 * x - 5, permute=True) == {
(-1, 1),
(-4, -1),
(1, -1),
(1, 1),
(-4, 1),
(-1, -1),
(4, 1),
(4, -1),
}
# test issue 18186
assert diophantine(
y ** 4 + x ** 4 - 2 ** 4 - 3 ** 4, syms=(x, y), permute=True
) == {(-3, -2), (-3, 2), (-2, -3), (-2, 3), (2, -3), (2, 3), (3, -2), (3, 2)}
assert diophantine(
y ** 4 + x ** 4 - 2 ** 4 - 3 ** 4, syms=(y, x), permute=True
) == {(-3, -2), (-3, 2), (-2, -3), (-2, 3), (2, -3), (2, 3), (3, -2), (3, 2)}
# issue 18122
assert check_solutions(x ** 2 - y)
assert check_solutions(y ** 2 - x)
assert diophantine((x ** 2 - y), t) == {(t, t ** 2)}
assert diophantine((y ** 2 - x), t) == {(t ** 2, -t)}
def test_diophantine():
assert check_solutions((x - y)*(y - z)*(z - x))
assert check_solutions((x - y)*(x**2 + y**2 - z**2))
assert check_solutions((x - 3*y + 7*z)*(x**2 + y**2 - z**2))
assert check_solutions((x**2 - 3*y**2 - 1))
assert check_solutions(y**2 + 7*x*y)
assert check_solutions(x**2 - 3*x*y + y**2)
assert check_solutions(z*(x**2 - y**2 - 15))
assert check_solutions(x*(2*y - 2*z + 5))
assert check_solutions((x**2 - 3*y**2 - 1)*(x**2 - y**2 - 15))
assert check_solutions((x**2 - 3*y**2 - 1)*(y - 7*z))
assert check_solutions((x**2 + y**2 - z**2)*(x - 7*y - 3*z + 4*w))
# Following test case caused problems in parametric representation
# But this can be solved by factroing out y.
# No need to use methods for ternary quadratic equations.
assert check_solutions(y**2 - 7*x*y + 4*y*z)
assert check_solutions(x**2 - 2*x + 1)
assert diophantine(x - y) == diophantine(Eq(x, y))
assert diophantine(3*x*pi - 2*y*pi) == set([(2*t_0, 3*t_0)])
eq = x**2 + y**2 + z**2 - 14
base_sol = set([(1, 2, 3)])
assert diophantine(eq) == base_sol
complete_soln = set(signed_permutations(base_sol.pop()))
assert diophantine(eq, permute=True) == complete_soln
assert diophantine(x**2 + 15*x/14 - 3) == set()
# test issue 11049
eq = 92*x**2 - 99*y**2 - z**2
coeff = eq.as_coefficients_dict()
assert _diop_ternary_quadratic_normal((x, y, z), coeff) == \
(9, 7, 51)
assert diophantine(eq) == set([(
891*p**2 + 9*q**2, -693*p**2 - 102*p*q + 7*q**2,
5049*p**2 - 1386*p*q - 51*q**2)])
eq = 2*x**2 + 2*y**2 - z**2
coeff = eq.as_coefficients_dict()
assert _diop_ternary_quadratic_normal((x, y, z), coeff) == \
(1, 1, 2)
assert diophantine(eq) == set([(
2*p**2 - q**2, -2*p**2 + 4*p*q - q**2,
4*p**2 - 4*p*q + 2*q**2)])
eq = 411*x**2+57*y**2-221*z**2
coeff = eq.as_coefficients_dict()
assert _diop_ternary_quadratic_normal((x, y, z), coeff) == \
(2021, 2645, 3066)
assert diophantine(eq) == \
set([(115197*p**2 - 446641*q**2, -150765*p**2 + 1355172*p*q -
584545*q**2, 174762*p**2 - 301530*p*q + 677586*q**2)])
eq = 573*x**2+267*y**2-984*z**2
coeff = eq.as_coefficients_dict()
assert _diop_ternary_quadratic_normal((x, y, z), coeff) == \
(49, 233, 127)
assert diophantine(eq) == \
set([(4361*p**2 - 16072*q**2, -20737*p**2 + 83312*p*q - 76424*q**2,
11303*p**2 - 41474*p*q + 41656*q**2)])
# this produces factors during reconstruction
eq = x**2 + 3*y**2 - 12*z**2
coeff = eq.as_coefficients_dict()
assert _diop_ternary_quadratic_normal((x, y, z), coeff) == \
(0, 2, 1)
assert diophantine(eq) == \
set([(24*p*q, 2*p**2 - 24*q**2, p**2 + 12*q**2)])
# solvers have not been written for every type
raises(NotImplementedError, lambda: diophantine(x*y**2 + 1))
# rational expressions
assert diophantine(1/x) == set()
assert diophantine(1/x + 1/y - S.Half)
set([(6, 3), (-2, 1), (4, 4), (1, -2), (3, 6)])
