def test_diop_general_sum_of_squares_quick():
for i in range(3, 10):
assert check_solutions(sum(i**2 for i in symbols(':%i' % i)) - i)
pytest.raises(ValueError, lambda: _diop_general_sum_of_squares((x, y), 2))
assert _diop_general_sum_of_squares((x, y, z), -2) == set()
eq = x**2 + y**2 + z**2 - (1 + 4 + 9)
assert diop_general_sum_of_squares(eq) == {(1, 2, 3)}
eq = u**2 + v**2 + x**2 + y**2 + z**2 - 1313
assert len(diop_general_sum_of_squares(eq, 3)) == 3
# issue sympy/sympy#11016
var = symbols(':5') + (symbols('6', negative=True),)
eq = Add(*[i**2 for i in var]) - 112
assert diophantine(eq) == {
(0, 1, 1, 5, 6, -7), (1, 1, 1, 3, 6, -8), (2, 3, 3, 4, 5, -7),
(0, 1, 1, 1, 3, -10), (0, 0, 4, 4, 4, -8), (1, 2, 3, 3, 5, -8),
(0, 1, 2, 3, 7, -7), (2, 2, 4, 4, 6, -6), (1, 1, 3, 4, 6, -7),
(0, 2, 3, 3, 3, -9), (0, 0, 2, 2, 2, -10), (1, 1, 2, 3, 4, -9),
(0, 1, 1, 2, 5, -9), (0, 0, 2, 6, 6, -6), (1, 3, 4, 5, 5, -6),
(0, 2, 2, 2, 6, -8), (0, 3, 3, 3, 6, -7), (0, 2, 3, 5, 5, -7),
(0, 1, 5, 5, 5, -6)}
# handle negated squares with signsimp
assert diophantine(12 - x**2 - y**2 - z**2) == {(2, 2, 2)}
# diophantine handles simplification, so classify_diop should
# not have to look for additional patterns that are removed
# by diophantine
eq = a**2 + b**2 + c**2 + d**2 - 4
pytest.raises(NotImplementedError, lambda: classify_diop(-eq))
def test_diop_general_sum_of_squares_quick():
for i in range(3, 10):
assert check_solutions(sum(i**2 for i in symbols(f':{i:d}')) - i)
pytest.raises(ValueError, lambda: _diop_general_sum_of_squares((x, y), 2))
assert _diop_general_sum_of_squares((x, y, z), -2) == set()
eq = x**2 + y**2 + z**2 - (1 + 4 + 9)
assert diop_general_sum_of_squares(eq) == {(1, 2, 3)}
eq = u**2 + v**2 + x**2 + y**2 + z**2 - 1313
assert len(diop_general_sum_of_squares(eq, 3)) == 3
# issue sympy/sympy#11016
var = symbols(':5') + (symbols('6', negative=True), )
eq = Add(*[i**2 for i in var]) - 112
assert diophantine(eq) == {(0, 1, 1, 5, 6, -7), (1, 1, 1, 3, 6, -8),
(2, 3, 3, 4, 5, -7), (0, 1, 1, 1, 3, -10),
(0, 0, 4, 4, 4, -8), (1, 2, 3, 3, 5, -8),
(0, 1, 2, 3, 7, -7), (2, 2, 4, 4, 6, -6),
(1, 1, 3, 4, 6, -7), (0, 2, 3, 3, 3, -9),
(0, 0, 2, 2, 2, -10), (1, 1, 2, 3, 4, -9),
(0, 1, 1, 2, 5, -9), (0, 0, 2, 6, 6, -6),
(1, 3, 4, 5, 5, -6), (0, 2, 2, 2, 6, -8),
(0, 3, 3, 3, 6, -7), (0, 2, 3, 5, 5, -7),
(0, 1, 5, 5, 5, -6)}
# handle negated squares with signsimp
assert diophantine(12 - x**2 - y**2 - z**2) == {(2, 2, 2)}
# diophantine handles simplification, so classify_diop should
# not have to look for additional patterns that are removed
# by diophantine
eq = a**2 + b**2 + c**2 + d**2 - 4
pytest.raises(NotImplementedError, lambda: classify_diop(-eq))
def test_diopcoverage():
eq = (2*x + y + 1)**2
assert diop_solve(eq) == {(t_0, -2*t_0 - 1)}
eq = 2*x**2 + 6*x*y + 12*x + 4*y**2 + 18*y + 18
assert diop_solve(eq) == {(t_0, -t_0 - 3), (2*t_0 - 3, -t_0)}
assert diop_quadratic(x + y**2 - 3) == {(-t**2 + 3, -t)}
assert diop_quadratic(x + y) is None # wrong type
assert diop_linear(x + y - 3) == (t_0, 3 - t_0)
assert diop_linear(x**2 - 1) is None # wrong type
assert base_solution_linear(0, 1, 2, t=None) == (0, 0)
ans = (3*t - 1, -2*t + 1)
assert base_solution_linear(4, 8, 12, t) == ans
assert base_solution_linear(4, 8, 12, t=None) == tuple(_.subs({t: 0}) for _ in ans)
assert cornacchia(1, 1, 20) is None
assert cornacchia(1, 1, 5) == {(1, 2)}
assert cornacchia(1, 2, 17) == {(3, 2)}
assert cornacchia(2, 3, 31) == set()
pytest.raises(ValueError, lambda: reconstruct(4, 20, 1))
assert gaussian_reduce(4, 1, 3) == (1, 1)
eq = -w**2 - x**2 - y**2 + z**2
assert (diop_general_pythagorean(eq) == diop_general_pythagorean(-eq) ==
(m1**2 + m2**2 - m3**2, 2*m1*m3, 2*m2*m3, m1**2 + m2**2 + m3**2))
assert check_param(Integer(3) + x/3, Integer(4) + x/2,
Integer(2), x) == (None, None)
assert check_param(Rational(3, 2), Integer(4) + x,
Integer(2), x) == (None, None)
assert check_param(Integer(4) + x, Rational(3, 2),
Integer(2), x) == (None, None)
assert _nint_or_floor(16, 10) == 2
assert _odd(1) == (not _even(1)) is True
assert _odd(0) == (not _even(0)) is False
assert _remove_gcd(2, 4, 6) == (1, 2, 3)
pytest.raises(TypeError, lambda: _remove_gcd((2, 4, 6)))
assert sqf_normal(2 * 3**2 * 5, 2 * 5 * 11, 2 * 7**2 * 11) == (11, 1, 5)
# it's ok if these pass some day when the solvers are implemented
pytest.raises(NotImplementedError, lambda: diophantine(x**2 + y**2 + x*y + 2*y*z - 12))
pytest.raises(NotImplementedError, lambda: diophantine(x**3 + y**2))
# issue sympy/sympy#11026
pytest.raises(NotImplementedError, lambda: diophantine(x**3 + y**3 - 2))
assert transformation_to_DN(x + y) is None # wrong type
assert find_DN(x + y) is None # wrong type
assert diop_ternary_quadratic(x + y) is None # wrong type
assert transformation_to_normal(x + y) is None # wrong type
assert parametrize_ternary_quadratic(x + y) is None # wrong type
assert diop_general_pythagorean(x + y) is None # wrong type
assert diop_general_sum_of_squares(x + y) is None # wrong type
assert diop_general_sum_of_even_powers(x + y) is None # wrong type
def _intersect(self, other):
from diofant import Dummy
from diofant.solvers.diophantine import diophantine
from diofant.sets.sets import imageset
if self.base_set is S.Integers:
if isinstance(other, ImageSet) and other.base_set is S.Integers:
f, g = self.lamda.expr, other.lamda.expr
n, m = self.lamda.variables[0], other.lamda.variables[0]
# Diophantine sorts the solutions according to the alphabetic
# order of the variable names, since the result should not depend
# on the variable name, they are replaced by the dummy variables
# below
a, b = Dummy('a'), Dummy('b')
f, g = f.subs(n, a), g.subs(m, b)
solns_set = diophantine(f - g)
if solns_set == set():
return EmptySet()
solns = list(diophantine(f - g))
if len(solns) == 1:
t = list(solns[0][0].free_symbols)[0]
else:
return
# since 'a' < 'b'
return imageset(Lambda(t, f.subs(a, solns[0][0])), S.Integers)
if other == S.Reals:
from diofant.solvers.diophantine import diophantine
from diofant.core.function import expand_complex
if len(self.lamda.variables
) > 1 or self.base_set is not S.Integers:
return
f = self.lamda.expr
n = self.lamda.variables[0]
n_ = Dummy(n.name, integer=True)
f_ = f.subs(n, n_)
re, im = f_.as_real_imag()
im = expand_complex(im)
sols = list(diophantine(im, n_))
if not sols:
return S.EmptySet
elif all(s[0].has(n_) is False for s in sols):
s = FiniteSet(*[s[0] for s in sols])
elif len(sols) == 1 and sols[0][0].has(n_):
s = imageset(Lambda(n_, sols[0][0]), S.Integers)
else:
return
return imageset(Lambda(n_, re), self.base_set.intersect(s))
def test_assumptions():
"""
Test whether diophantine respects the assumptions.
"""
# Test case taken from the below so question regarding assumptions in diophantine module
# http://stackoverflow.com/questions/23301941/how-can-i-declare-natural-symbols-with-sympy
m, n = symbols('m n', integer=True, positive=True)
diof = diophantine(n ** 2 + m * n - 500)
assert diof == {(5, 20), (40, 10), (95, 5), (121, 4), (248, 2), (499, 1)}
a, b = symbols('a b', integer=True, positive=False)
diof = diophantine(a*b + 2*a + 3*b - 6)
assert diof == {(-15, -3), (-9, -4), (-7, -5), (-6, -6), (-5, -8), (-4, -14)}
def test_assumptions():
"""
Test whether diophantine respects the assumptions.
"""
# Test case taken from the below so question regarding assumptions in diophantine module
# https//stackoverflow.com/questions/23301941/how-can-i-declare-natural-symbols-with-sympy
m, n = symbols('m n', integer=True, positive=True)
diof = diophantine(n ** 2 + m * n - 500)
assert diof == {(5, 20), (40, 10), (95, 5), (121, 4), (248, 2), (499, 1)}
a, b = symbols('a b', integer=True, positive=False)
diof = diophantine(a*b + 2*a + 3*b - 6)
assert diof == {(-15, -3), (-9, -4), (-7, -5), (-6, -6), (-5, -8), (-4, -14)}
def check_solutions(eq):
"""
Determines whether solutions returned by diophantine() satisfy the original
equation. Hope to generalize this so we can remove functions like check_ternay_quadratic,
check_solutions_normal, check_solutions()
"""
s = diophantine(eq)
terms = factor_list(eq)[1]
var = list(eq.free_symbols)
var.sort(key=default_sort_key)
okay = True
while len(s) and okay:
solution = s.pop()
okay = False
for term in terms:
subeq = term[0]
if simplify(_mexpand(Subs(subeq, var, solution).doit())) == 0:
okay = True
break
return okay
def test_quadratic_simple_hyperbolic_case():
# Simple Hyperbolic case: A = C = 0 and B != 0
assert diop_solve(3*x*y + 34*x - 12*y + 1) == {(-133, -11), (5, -57)}
assert diop_solve(6*x*y + 2*x + 3*y + 1) == set()
assert diop_solve(-13*x*y + 2*x - 4*y - 54) == {(27, 0)}
assert diop_solve(-27*x*y - 30*x - 12*y - 54) == {(-14, -1)}
assert diop_solve(2*x*y + 5*x + 56*y + 7) == {(-161, -3),
(-47, -6), (-35, -12), (-29, -69),
(-27, 64), (-21, 7), (-9, 1),
(105, -2)}
assert diop_solve(6*x*y + 9*x + 2*y + 3) == set()
assert diop_solve(x*y + x + y + 1) == {(-1, t), (t, -1)}
assert diophantine(48*x*y)
def test_diop_sum_of_even_powers():
eq = u**2 + v**2 + x**2 + y**2 + z**2 - 123
ans = diop_general_sum_of_squares(eq, oo) # allow oo to be used
assert len(ans) == 14
eq = x**4 + y**4 + z**4 - 2673
assert diop_solve(eq) == {(3, 6, 6), (2, 4, 7)}
assert diop_general_sum_of_even_powers(eq, 2) == {(3, 6, 6), (2, 4, 7)}
pytest.raises(NotImplementedError, lambda: diop_general_sum_of_even_powers(-eq, 2))
neg = symbols('neg', negative=True)
eq = x**4 + y**4 + neg**4 - 2673
assert diop_general_sum_of_even_powers(eq) == {(-3, 6, 6)}
assert diophantine(x**4 + y**4 + 2) == set()
assert diop_general_sum_of_even_powers(x**4 + y**4 - 2, limit=0) == set()
def test_quadratic_simple_hyperbolic_case():
# Simple Hyperbolic case: A = C = 0 and B != 0
assert diop_solve(3*x*y + 34*x - 12*y + 1) == \
{(-Integer(133), -Integer(11)), (Integer(5), -Integer(57))}
assert diop_solve(6*x*y + 2*x + 3*y + 1) == set()
assert diop_solve(-13*x*y + 2*x - 4*y - 54) == {(Integer(27), Integer(0))}
assert diop_solve(-27*x*y - 30*x - 12*y - 54) == {(-Integer(14), -Integer(1))}
assert diop_solve(2*x*y + 5*x + 56*y + 7) == {(-Integer(161), -Integer(3)),
(-Integer(47), -Integer(6)), (-Integer(35), -Integer(12)), (-Integer(29), -Integer(69)),
(-Integer(27), Integer(64)), (-Integer(21), Integer(7)), (-Integer(9), Integer(1)),
(Integer(105), -Integer(2))}
assert diop_solve(6*x*y + 9*x + 2*y + 3) == set()
assert diop_solve(x*y + x + y + 1) == {(-Integer(1), t), (t, -Integer(1))}
assert diophantine(48*x*y)
def test_quadratic_simple_hyperbolic_case():
# Simple Hyperbolic case: A = C = 0 and B != 0
assert diop_solve(3 * x * y + 34 * x - 12 * y + 1) == {(-133, -11),
(5, -57)}
assert diop_solve(6 * x * y + 2 * x + 3 * y + 1) == set()
assert diop_solve(-13 * x * y + 2 * x - 4 * y - 54) == {(27, 0)}
assert diop_solve(-27 * x * y - 30 * x - 12 * y - 54) == {(-14, -1)}
assert diop_solve(2 * x * y + 5 * x + 56 * y + 7) == {
(-161, -3), (-47, -6), (-35, -12), (-29, -69), (-27, 64), (-21, 7),
(-9, 1), (105, -2)
}
assert diop_solve(6 * x * y + 9 * x + 2 * y + 3) == set()
assert diop_solve(x * y + x + y + 1) == {(-1, t), (t, -1)}
assert diophantine(48 * x * y)
def test_diop_sum_of_even_powers():
eq = u**2 + v**2 + x**2 + y**2 + z**2 - 123
ans = diop_general_sum_of_squares(eq, oo) # allow oo to be used
assert len(ans) == 14
eq = x**4 + y**4 + z**4 - 2673
assert diop_solve(eq) == {(3, 6, 6), (2, 4, 7)}
assert diop_general_sum_of_even_powers(eq, 2) == {(3, 6, 6), (2, 4, 7)}
pytest.raises(NotImplementedError,
lambda: diop_general_sum_of_even_powers(-eq, 2))
neg = symbols('neg', negative=True)
eq = x**4 + y**4 + neg**4 - 2673
assert diop_general_sum_of_even_powers(eq) == {(-3, 6, 6)}
assert diophantine(x**4 + y**4 + 2) == set()
assert diop_general_sum_of_even_powers(x**4 + y**4 - 2, limit=0) == set()
def check_integrality(eq):
"""
Check that the solutions returned by diophantine() are integers.
This should be seldom needed except for general quadratic
equations which are solved with rational transformations.
"""
def _check_values(x):
""" Check a number of values. """
for i in range(-4, 4):
if not isinstance(simplify(x.subs(t, i)), Integer):
return False
return True
for soln in diophantine(eq, param=t):
for x in soln:
if not _check_values(x):
return False
return True
def check_solutions(eq):
"""
Determines whether solutions returned by diophantine() satisfy the original
equation. Hope to generalize this so we can remove functions like check_ternay_quadratic,
check_solutions_normal, check_solutions()
"""
s = diophantine(eq)
factors = Mul.make_args(eq)
var = list(eq.free_symbols)
var.sort(key=default_sort_key)
while s:
solution = s.pop()
for f in factors:
if diop_simplify(f.subs(zip(var, solution))) == 0:
break
else:
return False
return True
def check_solutions(eq):
"""
Determines whether solutions returned by diophantine() satisfy the original
equation. Hope to generalize this so we can remove functions like check_ternay_quadratic,
check_solutions_normal, check_solutions()
"""
s = diophantine(eq)
factors = Mul.make_args(eq)
var = list(eq.free_symbols)
var.sort(key=default_sort_key)
while s:
solution = s.pop()
for f in factors:
if diop_simplify(f.subs(zip(var, solution))) == 0:
break
else:
return False
return True
def test_sympyissue_11959():
solution = {(11, -71), (33, -64), (49, -53), (54, -48), (-59, -41),
(-61, -38), (65, -32), (-70, -17), (72, -10), (-72, 6),
(-68, 25), (42, 60), (39, 62), (-24, 69), (18, 71), (-5, 73)}
assert diophantine(10 * x**2 - 6 * x + 10 * y**2 - 14 * y -
52548) == solution
def test_sympyissue_9539():
assert diophantine(6*w + 9*y + 20*x - z) == {(t_0, t_1, t_1 + t_2,
6*t_0 + 29*t_1 + 9*t_2)}
def test_sympyissue_8943():
assert diophantine(
(3 * (x**2 + y**2 + z**2) - 14 * (x * y + y * z + z * x))) == {(0, 0,
0)}
def test_sympyissue_9539():
assert diophantine(6 * w + 9 * y + 20 * x - z) == {
(t_0, t_1, t_1 + t_2, 6 * t_0 + 29 * t_1 + 9 * t_2)
}
def test_diopcoverage():
eq = (2 * x + y + 1)**2
assert diop_solve(eq) == {(t_0, -2 * t_0 - 1)}
eq = 2 * x**2 + 6 * x * y + 12 * x + 4 * y**2 + 18 * y + 18
assert diop_solve(eq) == {(t_0, -t_0 - 3), (2 * t_0 - 3, -t_0)}
assert diop_quadratic(x + y**2 - 3) == {(-t**2 + 3, -t)}
assert diop_quadratic(x + y) is None # wrong type
assert diop_linear(x + y - 3) == (t_0, 3 - t_0)
assert diop_linear(x**2 - 1) is None # wrong type
assert base_solution_linear(0, 1, 2, t=None) == (0, 0)
ans = (3 * t - 1, -2 * t + 1)
assert base_solution_linear(4, 8, 12, t) == ans
assert base_solution_linear(4, 8, 12,
t=None) == tuple(_.subs({t: 0}) for _ in ans)
assert cornacchia(1, 1, 20) is None
assert cornacchia(1, 1, 5) == {(2, 1)}
assert cornacchia(1, 2, 17) == {(3, 2)}
assert cornacchia(2, 3, 31) == set()
assert cornacchia(1, 4, 52) == {(4, 3)}
pytest.raises(ValueError, lambda: reconstruct(4, 20, 1))
assert gaussian_reduce(4, 1, 3) == (1, 1)
eq = -w**2 - x**2 - y**2 + z**2
assert (diop_general_pythagorean(eq) == diop_general_pythagorean(-eq) ==
(m1**2 + m2**2 - m3**2, 2 * m1 * m3, 2 * m2 * m3,
m1**2 + m2**2 + m3**2))
assert check_param(Integer(3) + x / 3,
Integer(4) + x / 2, Integer(2), x) == (None, None)
assert check_param(Rational(3, 2),
Integer(4) + x, Integer(2), x) == (None, None)
assert check_param(Integer(4) + x, Rational(3, 2), Integer(2),
x) == (None, None)
assert _nint_or_floor(16, 10) == 2
assert _odd(1) == (not _even(1)) is True
assert _odd(0) == (not _even(0)) is False
assert _remove_gcd(2, 4, 6) == (1, 2, 3)
assert sqf_normal(2 * 3**2 * 5, 2 * 5 * 11, 2 * 7**2 * 11) == (11, 1, 5)
# it's ok if these pass some day when the solvers are implemented
pytest.raises(NotImplementedError,
lambda: diophantine(x**2 + y**2 + x * y + 2 * y * z - 12))
pytest.raises(NotImplementedError, lambda: diophantine(x**3 + y**2))
# issue sympy/sympy#11026
pytest.raises(NotImplementedError, lambda: diophantine(x**3 + y**3 - 2))
assert transformation_to_DN(x + y) is None # wrong type
assert find_DN(x + y) is None # wrong type
assert diop_ternary_quadratic(x + y) is None # wrong type
assert transformation_to_normal(x + y) is None # wrong type
assert parametrize_ternary_quadratic(x + y) is None # wrong type
assert diop_general_pythagorean(x + y) is None # wrong type
assert diop_general_sum_of_squares(x + y) is None # wrong type
assert diop_general_sum_of_even_powers(x + y) is None # wrong type
assert diop_quadratic(x**2 + y**2 - 1**2 - 3**4) == \
{(-9, -1), (-9, 1), (-1, -9), (-1, 9), (1, -9), (1, 9), (9, -1), (9, 1)}
def test_diophantine():
# Commented out test cases should be uncommented after
# the bug with factor_list() gets merged.
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(Integer(0)) == {(t, )}
assert diophantine(x - y) == diophantine(Eq(x, y))
assert diophantine(3 * x * pi - 2 * y * pi) == {(2 * t_0, 3 * t_0)}
assert diophantine(x**2 + y**2 + z**2 - 14) == {(1, 2, 3)}
assert diophantine(x**2 + 15 * x / 14 - 3) == set()
# test issue sympy/sympy#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
pytest.raises(NotImplementedError, lambda: diophantine(x * y**2 + 1))
# rational expressions
assert diophantine(1 / x) == set()
assert diophantine(1 / x + 1 / y - Rational(1, 2)) == {(6, 3), (-2, 1),
(4, 4), (1, -2),
(3, 6)}
# issue sympy/sympy#9538
eq = x - 3 * y + 2
assert diophantine(eq, syms=[y, x]) == {(t_0, 3 * t_0 - 2)}
assert diophantine(eq, syms=[x, y]) == {(3 * t_0 - 2, t_0)}
pytest.raises(TypeError, lambda: diophantine(eq, syms={y, x}))
def test_sympyissue_9106():
eq = -48 - 2*x*(3*x - 1) + y*(3*y - 1)
v = (x, y)
for sol in diophantine(eq):
assert not diop_simplify(eq.xreplace(dict(zip(v, sol))))
def test_sympyissue_9106():
eq = -48 - 2 * x * (3 * x - 1) + y * (3 * y - 1)
v = (x, y)
for sol in diophantine(eq):
assert not diop_simplify(eq.xreplace(dict(zip(v, sol))))
def test_input_format():
pytest.raises(TypeError, lambda: diophantine(sin(x)))
pytest.raises(TypeError, lambda: diophantine(3))
pytest.raises(TypeError, lambda: diophantine(x / pi - 3))
def test_diophantine():
# Commented out test cases should be uncommented after
# the bug with factor_list() gets merged.
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(Integer(0)) == {(t,)}
assert diophantine(x - y) == diophantine(Eq(x, y))
assert diophantine(3*x*pi - 2*y*pi) == {(2*t_0, 3*t_0)}
assert diophantine(x**2 + y**2 + z**2 - 14) == {(1, 2, 3)}
assert diophantine(x**2 + 15*x/14 - 3) == set()
# test issue sympy/sympy#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
pytest.raises(NotImplementedError, lambda: diophantine(x*y**2 + 1))
# rational expressions
assert diophantine(1/x) == set()
assert diophantine(1/x + 1/y - Rational(1, 2)) == {(6, 3), (-2, 1),
(4, 4), (1, -2), (3, 6)}
def test_sympyissue_9538():
eq = x - 3 * y + 2
assert diophantine(eq, syms=[y, x]) == {(t_0, 3 * t_0 - 2)}
pytest.raises(TypeError, lambda: diophantine(eq, syms={y, x}))
assert diophantine(eq, syms=[x, y]) == {(3 * t_0 - 2, t_0)}
def test_sympyissue_8943():
assert diophantine((3*(x**2 + y**2 + z**2) -
14*(x*y + y*z + z*x))) == {(0, 0, 0)}
def test_sympyissue_9538():
eq = x - 3*y + 2
assert diophantine(eq, syms=[y, x]) == {(t_0, 3*t_0 - 2)}
pytest.raises(TypeError, lambda: diophantine(eq, syms={y, x}))
assert diophantine(eq, syms=[x, y]) == {(3*t_0 - 2, t_0)}
def test_input_format():
pytest.raises(TypeError, lambda: diophantine(sin(x)))
pytest.raises(TypeError, lambda: diophantine(3))
pytest.raises(TypeError, lambda: diophantine(x/pi - 3))