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"""

Problem 44: https://projecteuler.net/problem=44

Pentagonal numbers are generated by the formula, Pn=n(3n−1)/2. The first ten

pentagonal numbers are:

1, 5, 12, 22, 35, 51, 70, 92, 117, 145, ...

It can be seen that P4 + P7 = 22 + 70 = 92 = P8. However, their difference,

70 − 22 = 48, is not pentagonal.

Find the pair of pentagonal numbers, Pj and Pk, for which their sum and difference

are pentagonal and D = |Pk − Pj| is minimised; what is the value of D?

"""

def is_pentagonal(n: int) -> bool:

"""

Returns True if n is pentagonal, False otherwise.

>>> is_pentagonal(330)

True

>>> is_pentagonal(7683)

False

>>> is_pentagonal(2380)

True

"""

root = (1 + 24 * n) ** 0.5

return ((1 + root) / 6) % 1 == 0

def solution(limit: int = 5000) -> int:

"""

Returns the minimum difference of two pentagonal numbers P1 and P2 such that

P1 + P2 is pentagonal and P2 - P1 is pentagonal.

>>> solution(5000)

5482660

"""

pentagonal_nums = [(i * (3 * i - 1)) // 2 for i in range(1, limit)]

for i, pentagonal_i in enumerate(pentagonal_nums):

for j in range(i, len(pentagonal_nums)):

pentagonal_j = pentagonal_nums[j]

a = pentagonal_i + pentagonal_j

b = pentagonal_j - pentagonal_i

if is_pentagonal(a) and is_pentagonal(b):

return b

return -1

if __name__ == "__main__":

print(f"{solution() = }")

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