## Pentagon numbers

### Problem 44

Published on Friday, 23rd May 2003, 06:00 pm; Solved by 33798; Difficulty rating: 5%Pentagonal numbers are generated by the formula, P_{n}=`n`(3`n`−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 are pentagonal and D = |P_{k} − P_{j}| is minimised; what is the value of D?