def test_diopcoverage():
eq = (2 * x + y + 1)**2
assert diop_solve(eq) == set([(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) == set([(t_0, -t_0 - 3), (2 * t_0 - 3, -t_0)])
assert diop_quadratic(x + y**2 - 3) == set([(-t**2 + 3, -t)])
assert diop_linear(x + y - 3) == (t_0, 3 - t_0)
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) == set([(2, 1)])
assert cornacchia(1, 2, 17) == set([(3, 2)])
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(S(3) + x / 3, S(4) + x / 2, S(2), x) == (None, None)
assert check_param(S(3) / 2, S(4) + x, S(2), x) == (None, None)
assert check_param(S(4) + x, S(3) / 2, S(2), x) == (None, None)
assert _nint_or_floor(16, 10) == 2
assert _odd(1) == (not _even(1)) == True
assert _odd(0) == (not _even(0)) == False
assert _remove_gcd(2, 4, 6) == (1, 2, 3)
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
raises(NotImplementedError,
lambda: diophantine(x**2 + y**2 + x * y + 2 * y * z - 12))
raises(NotImplementedError, lambda: diophantine(x**3 + y**2))
assert diop_quadratic(x**2 + y**2 - 1**2 - 3**4) == \
set([(-9, -1), (-9, 1), (-1, -9), (-1, 9), (1, -9), (1, 9), (9, -1), (9, 1)])
def test_diopcoverage():
eq = (2*x + y + 1)**2
assert diop_solve(eq) == set([(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) == set([(t_0, -t_0 - 3), (2*t_0 - 3, -t_0)])
assert diop_quadratic(x + y**2 - 3) == set([(-t**2 + 3, -t)])
assert diop_linear(x + y - 3) == (t_0, 3 - t_0)
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) == set([(2, 1)])
assert cornacchia(1, 2, 17) == set([(3, 2)])
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(S(3) + x/3, S(4) + x/2, S(2), x) == (None, None)
assert check_param(S(3)/2, S(4) + x, S(2), x) == (None, None)
assert check_param(S(4) + x, S(3)/2, S(2), x) == (None, None)
assert _nint_or_floor(16, 10) == 2
assert _odd(1) == (not _even(1)) == True
assert _odd(0) == (not _even(0)) == False
assert _remove_gcd(2, 4, 6) == (1, 2, 3)
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
raises(NotImplementedError, lambda: diophantine(x**2 + y**2 + x*y + 2*y*z - 12))
raises(NotImplementedError, lambda: diophantine(x**3 + y**2))
assert diop_quadratic(x**2 + y**2 - 1**2 - 3**4) == \
set([(-9, -1), (-9, 1), (-1, -9), (-1, 9), (1, -9), (1, 9), (9, -1), (9, 1)])
nums.append(num*subprimes[ind])
GenNum(num, ind+1)
num *= subprimes[ind]
# Generate all prime numbers below 150
primes = list(sp.sieve.primerange(1, 150))
subprimes = []
for p in primes:
if p%4 == 1: # Take only prime numbers of 4*k+1
subprimes.append(p)
exp = [1 for i in range(len(subprimes))]
GenNum(1, 0)
nums = np.unique(np.sort(nums))
for n in nums:
sum += np.sum(np.unique(np.sort(list(cornacchia(1,1,n)), axis = 1), axis = 0)[:,0]) # Find possible a and b that sum of squares = n
print(n, sum)
print("Ans: ", sum)
# Appendix: Partial Sums of the last few values of n
"""
************************************************
* n * Partial Sum of S(n) *
************************************************
2650349212148801960854945 438361883167503705
2791544286713887969868305 443465771270435289
3171729385030533494137885 448906838435992373
3376082419233843034699789 454519564158817836
3608581638922830790317565 460323377560668066
3650480990318161191366245 466159827222344194