Problem044.py

"""
Project Euler Problem 44
========================

Pentagonal numbers are generated by the formula, P[n]=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 P[4] + P[7] = 22 + 70 = 92 = P[8]. However, their
difference, 70 - 22 = 48, is not pentagonal.

Find the pair of pentagonal numbers, P[j] and P[k], for which their sum
and difference is pentagonal and D = |P[k] - P[j]| is minimised; what is
the value of D?
"""
from math import sqrt, floor



def P_n(n):
    return n*(3*n-1) // 2

def is_pentagonal(Pn):
    """ Solving the quadratic expression we can determine n and see is integer """
    n =  (1/2 + sqrt(1/4+4*3/2*Pn))/3
    return floor(n) == n


found = False
n = 0
pentagons = set()
difference = {}
while not found:
    n += 1
    new_pentagon = P_n(n)

    for pentagon in pentagons:
        if new_pentagon - pentagon in pentagons and is_pentagonal(new_pentagon+pentagon):
            print(new_pentagon-pentagon)
            found = True
            break
            
    pentagons.add(new_pentagon)