def test_ternary_quadratic():
# solution with 3 parameters
s = diophantine(2 * x**2 + y**2 - 2 * z**2)
p, q, r = ordered(S(s).free_symbols)
assert s == {
(p**2 - 2 * q**2, -2 * p**2 + 4 * p * q - 4 * p * r - 4 * q**2,
p**2 - 4 * p * q + 2 * q**2 - 4 * q * r)
}
# solution with Mul in solution
s = diophantine(x**2 + 2 * y**2 - 2 * z**2)
assert s == {(4 * p * q, p**2 - 2 * q**2, p**2 + 2 * q**2)}
# solution with no Mul in solution
s = diophantine(2 * x**2 + 2 * y**2 - z**2)
assert s == {(2 * p**2 - q**2, -2 * p**2 + 4 * p * q - q**2,
4 * p**2 - 4 * p * q + 2 * q**2)}
# reduced form when parametrized
s = diophantine(3 * x**2 + 72 * y**2 - 27 * z**2)
assert s == {(24 * p**2 - 9 * q**2, 6 * p * q, 8 * p**2 + 3 * q**2)}
assert parametrize_ternary_quadratic(3 * x**2 + 2 * y**2 - z**2 -
2 * x * y + 5 * y * z -
7 * y * z) == (2 * p**2 - 2 * p * q -
q**2, 2 * p**2 +
2 * p * q - q**2,
2 * p**2 - 2 * p * q +
3 * q**2)
assert parametrize_ternary_quadratic(124 * x**2 - 30 * y**2 -
7729 * z**2) == (
-1410 * p**2 - 363263 * q**2,
2700 * p**2 + 30916 * p * q -
695610 * q**2, -60 * p**2 +
5400 * p * q + 15458 * q**2)
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)
