Problem 799

Pentagonal Puzzle

Pentagonal numbers are generated by the formula: $P_{n} = \frac{1}{2} n (3 n - 1)$ giving the sequence:

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

Some pentagonal numbers can be expressed as the sum of two other pentagonal numbers.
For example:

$P_{8} = 92 = 22 + 70 = P_{4} + P_{7}$

$3577$ is the smallest pentagonal number that can be expressed as the sum of two pentagonal numbers in two different ways:

$\begin{aligned} P_{49} = 3577 & = 3432 + 145 = P_{48} + P_{10} \\ = 3290 + 287 = P_{47} + P_{14} \end{aligned}$

$107602$ is the smallest pentagonal number that can be expressed as the sum of two pentagonal numbers in three different ways.

Find the smallest pentagonal number that can be expressed as the sum of two pentagonal numbers in over $100$ different ways.

五边形数由公式 $P_{n} = \frac{1}{2} n (3 n - 1)$ 生成：

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

有些五边形数可以表示成另外两个五边形数之和。
例如：

$P_{8} = 92 = 22 + 70 = P_{4} + P_{7}$

$3577$ 是最小的可以用两种不同方式表示成两个五边形数之和的五边形数：

$\begin{aligned} P_{49} = 3577 & = 3432 + 145 = P_{48} + P_{10} \\ = 3290 + 287 = P_{47} + P_{14} \end{aligned}$

$107602$ 是最小的可以用三种不同方式表示成两个五边形数之和的五边形数。

求最小的可以用超过 $100$ 种不同方式表示成两个五边形数之和的五边形数。