Problem 44

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 $D$ ?

五边形数

五边形数由公式 $P_{n} = n (3 n - 1) / 2$ 给出。前十个五边形数是：

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

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？