p44.py

"""
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?
"""

import math

def main():
	findMinimalDifference()

def findMinimalDifference():
	#pentagonal_nums initialized with i = 1 and P(1) = 1.
	pentagonal_nums = [1]
	n=1
	minimal_pentagonal_difference = 1
	while True:
		n+=1
		pentagonal_nums.append(int(n/2*(3*n - 1)))
		for i in range(len(pentagonal_nums)-1, 0, -1):
			if (checkForPentagonalNumber(abs(pentagonal_nums[i]+pentagonal_nums[-1]))) and (checkForPentagonalNumber(abs(pentagonal_nums[i]-pentagonal_nums[-1]))):
				minimal_pentagonal_difference = abs(pentagonal_nums[i]-pentagonal_nums[-1])
				print("Pj = ", n, "\nPk = ", i, "\nD = |Pk-Pj| = ", minimal_pentagonal_difference)
				return minimal_pentagonal_difference


def checkForPentagonalNumber(input):
	# using the equation to calculate an index from a possible Pentagonal number, see: https://en.wikipedia.org/wiki/Pentagonal_number#Tests_for_pentagonal_numbers

	index = 1/6*(math.sqrt(24*input+1)+1)
	return (index % 1 == 0 and index>=1) # check if index is a whole number AND is >=1 (i.e. is a natural number)

if __name__ == "__main__":
	main()