If \(p_D(n)\) denotes the number of partitions of n into distinct parts, it is known that for \(\alpha \ge 1\) and \(n\ge 0\),$$\begin{aligned} p_D\left( 5^{2\alpha +1}n+\frac{5^{2\alpha +2}-1}{24}\right) \equiv 0\pmod {5^\alpha }. \end{aligned}$$We give a completely elementary proof of this fact.

中文翻译：

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划分为5的不同部分模幂

如果\（p_D（n）\）表示n划分为不同部分的数量，则对于\（\ alpha \ ge 1 \）和\（n \ ge 0 \），$$ \ begin {aligned } p_D \ left（5 ^ {2 \ alpha +1} n + \ frac {5 ^ {2 \ alpha +2} -1} {24} \ right）\ equiv 0 \ pmod {5 ^ \ alpha}。\ end {aligned} $$我们提供了这一事实的基本证明。