Pentagon Numbers

Pentagonal numbers are generated by the formula, $P_{n} = n (3 n - 1) / 2$ . The first ten pentagonal numbers are: $1, 5, 12, 22, 35, 51, 70, 92, 117, 145, \dots$

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 $D$ ?

Write-up coming later

The complete problem is available here. An approach, code, and answer will be added later.