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)])