Problem 44

Pentagon numbers

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 P₄ + P₇ = 22 + 70 = 92 = P₈. 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?

五边形数

五边形数由公式P_n=n(3n−1)/2生成。前十个五边形数是：

1, 5, 12, 22, 35, 51, 70, 92, 117, 145, ...

可以看出P₄ + P₇ = 22 + 70 = 92 = P₈。然而，它们的差70 − 22 = 48并不是五边形数。

在所有和差均为五边形数的五边形数对P_j和P_k中，找出使D = |P_k − P_j|最小的一对；此时D的值是多少？