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_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) == \
(64*p**2 - 24*p*q, -64*p*q + 64*q**2, 40*p*q)
# this cannot be tested with diophantine because it will
# factor into a product
assert diop_solve(x*y + 2*y*z) == (-4*p*q, -2*n1*p**2 + 2*p**2, 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 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)])
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)])
assert diophantine(x**2 + y**2 + z**2 - 14) == set([(1, 2, 3)])
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)])
