0044-pentagonal-sum-difference.py

"""
Problem 44

Pentagonal numbers are generated by the formula, Pn=n(3n1)/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 is pentagonal and D = |Pk - Pj| is minimised; what is the value of D?
"""
import math
def is_pentagonal(x):
    n = (math.sqrt((24 * x) + 1) + 1) / 6

    if n == int(n) and n > 0:
        return True
    return False

def brute_force():
    start = 1
    n = 1
    pentagonal = lambda x: x * (3 * x - 1) / 2
    limit = 3000

    p_list = [pentagonal(n) for n in range(1, limit)]
    for i in range(start, len(p_list)):
        for j in range(i, len(p_list)):
            if is_pentagonal(p_list[i] + p_list[j]) and \
            is_pentagonal(abs(p_list[i] - p_list[j])):
                print p_list[i], p_list[j], p_list[i] - p_list[j]

print "Answer:", brute_force()