def decompose_prime_square(n):
"""
Attempt to write n as the sum of a prime and twice a square.
>>> decompose_prime_square(9)
(7, 1)
>>> decompose_prime_square(15)
(7, 2)
>>> decompose_prime_square(21)
(3, 3)
>>> decompose_prime_square(25)
(7, 3)
>>> decompose_prime_square(27)
(19, 2)
>>> decompose_prime_square(33)
(31, 1)
"""
for p in up_to(n, primes()):
if p == 2:
continue
residue = n - p
assert residue % 2 == 0, residue
square = residue // 2
if is_perfect_square(square):
return (p, isqrt(square))
def is_triangular_number(n):
"""
>>> all(is_triangular_number(n) for n in up_to(1000000000, figurate_numbers(3)))
True
>>> any(is_triangular_number(n + 1) for n in up_to(1000000000, figurate_numbers(3)))
False
"""
#http://en.wikipedia.org/wiki/Triangular_number
return is_perfect_square(8 * n + 1)
