Python/project_euler/problem_44/sol1.py

"""
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


if __name__ == "__main__":
    print(f"{solution() = }")