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)
assert diop_general_sum_of_squares(x**2 + y**2 - 2) is None
assert diop_general_sum_of_squares(x**2 + y**2 + z**2 + 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 11016
var = symbols(':5') + (symbols('6', negative=True), )
eq = Add(*[i**2 for i in var]) - 112
base_soln = {(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)}
assert diophantine(eq) == base_soln
assert len(diophantine(eq, permute=True)) == 196800
# 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
raises(NotImplementedError, lambda: classify_diop(-eq))
def test_sum_of_squares_powers():
tru = set([
(0, 0, 1, 1, 11), (0, 0, 5, 7, 7), (0, 1, 3, 7, 8), (0, 1, 4, 5, 9),
(0, 3, 4, 7, 7), (0, 3, 5, 5, 8), (1, 1, 2, 6, 9), (1, 1, 6, 6, 7),
(1, 2, 3, 3, 10), (1, 3, 4, 4, 9), (1, 5, 5, 6, 6), (2, 2, 3, 5, 9),
(2, 3, 5, 6, 7), (3, 3, 4, 5, 8)])
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
assert ans == tru
raises(ValueError, lambda: list(sum_of_squares(10, -1)))
assert list(sum_of_squares(-10, 2)) == []
assert list(sum_of_squares(2, 3)) == []
assert list(sum_of_squares(0, 3, True)) == [(0, 0, 0)]
assert list(sum_of_squares(0, 3)) == []
assert list(sum_of_squares(4, 1)) == [(2,)]
assert list(sum_of_squares(5, 1)) == []
assert list(sum_of_squares(50, 2)) == [(5, 5), (1, 7)]
assert list(sum_of_squares(11, 5, True)) == [
(1, 1, 1, 2, 2), (0, 0, 1, 1, 3)]
assert list(sum_of_squares(8, 8)) == [(1, 1, 1, 1, 1, 1, 1, 1)]
assert [len(list(sum_of_squares(i, 5, True))) for i in range(30)] == [
1, 1, 1, 1, 2,
2, 1, 1, 2, 2,
2, 2, 2, 3, 2,
1, 3, 3, 3, 3,
4, 3, 3, 2, 2,
4, 4, 4, 4, 5]
assert [len(list(sum_of_squares(i, 5))) for i in range(30)] == [
0, 0, 0, 0, 0,
1, 0, 0, 1, 0,
0, 1, 0, 1, 1,
0, 1, 1, 0, 1,
2, 1, 1, 1, 1,
1, 1, 1, 1, 3]
for i in range(30):
s1 = set(sum_of_squares(i, 5, True))
assert not s1 or all(sum(j**2 for j in t) == i for t in s1)
s2 = set(sum_of_squares(i, 5))
assert all(sum(j**2 for j in t) == i for t in s2)
raises(ValueError, lambda: list(sum_of_powers(2, -1, 1)))
raises(ValueError, lambda: list(sum_of_powers(2, 1, -1)))
assert list(sum_of_powers(-2, 3, 2)) == [(-1, -1)]
assert list(sum_of_powers(-2, 4, 2)) == []
assert list(sum_of_powers(2, 1, 1)) == [(2,)]
assert list(sum_of_powers(2, 1, 3, True)) == [(0, 0, 2), (0, 1, 1)]
assert list(sum_of_powers(5, 1, 2, True)) == [(0, 5), (1, 4), (2, 3)]
assert list(sum_of_powers(6, 2, 2)) == []
assert list(sum_of_powers(3**5, 3, 1)) == []
assert list(sum_of_powers(3**6, 3, 1)) == [(9,)] and (9**3 == 3**6)
assert list(sum_of_powers(2**1000, 5, 2)) == []