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Problem 44

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Pentagon numbers

Pentagonal numbers are generated by the formula, $P_n=n(3n−1)/2$. The first ten pentagonal numbers are:

$$1, 5, 12, 22, 35, 51, 70, 92, 117, 145, \ldots $$

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 $D$?

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五边形数

五边形数由公式$P_n=n(3n−1)/2$给出。前十个五边形数是：

$$1, 5, 12, 22, 35, 51, 70, 92, 117, 145, \ldots $$

可以看出$P_4 + P_7 = 22 + 70 = 92 = P_8$。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？

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