def test_diop_ternary_quadratic():
assert check_solutions(2*x**2 + z**2 + y**2 - 4*x*y)
assert check_solutions(x**2 - y**2 - z**2 - x*y - y*z)
assert check_solutions(3*x**2 - x*y - y*z - x*z)
assert check_solutions(x**2 - y*z - x*z)
assert check_solutions(5*x**2 - 3*x*y - x*z)
assert check_solutions(4*x**2 - 5*y**2 - x*z)
assert check_solutions(3*x**2 + 2*y**2 - z**2 - 2*x*y + 5*y*z - 7*y*z)
assert check_solutions(8*x**2 - 12*y*z)
assert check_solutions(45*x**2 - 7*y**2 - 8*x*y - z**2)
assert check_solutions(x**2 - 49*y**2 - z**2 + 13*z*y -8*x*y)
assert check_solutions(90*x**2 + 3*y**2 + 5*x*y + 2*z*y + 5*x*z)
assert check_solutions(x**2 + 3*y**2 + z**2 - x*y - 17*y*z)
assert check_solutions(x**2 + 3*y**2 + z**2 - x*y - 16*y*z + 12*x*z)
assert check_solutions(x**2 + 3*y**2 + z**2 - 13*x*y - 16*y*z + 12*x*z)
assert check_solutions(x*y - 7*y*z + 13*x*z)
assert diop_ternary_quadratic_normal(x**2 + y**2 + z**2) == (None, None, None)
assert diop_ternary_quadratic_normal(x**2 + y**2) is None
raises(ValueError, lambda:
_diop_ternary_quadratic_normal((x, y, z),
{x*y: 1, x**2: 2, y**2: 3, z**2: 0}))
eq = -2*x*y - 6*x*z + 7*y**2 - 3*y*z + 4*z**2
assert diop_ternary_quadratic(eq) == (7, 2, 0)
assert diop_ternary_quadratic_normal(4*x**2 + 5*y**2 - z**2) == \
(1, 0, 2)
assert diop_ternary_quadratic(x*y + 2*y*z) == \
(-2, 0, n1)
eq = -5*x*y - 8*x*z - 3*y*z + 8*z**2
assert parametrize_ternary_quadratic(eq) == \
(8*p**2 - 3*p*q, -8*p*q + 8*q**2, 5*p*q)
# this cannot be tested with diophantine because it will
# factor into a product
assert diop_solve(x*y + 2*y*z) == (-2*p*q, -n1*p**2 + p**2, p*q)
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)}
