关于拉马努金的一个奇特恒等式
数论
2020-01-07 v2
摘要
拉马努金写下了如下恒等式 \begin{align*} \sqrt{2 \left(1 - \frac{1}{3^2}\right) \left(1 - \frac{1}{7^2}\right) \left(1 - \frac{1}{11^2}\right) \left(1 - \frac{1}{19^2}\right)} \ = \ \left(1 + \frac{1}{7}\right) \left(1 + \frac{1}{11}\right) \left(1 + \frac{1}{19}\right). \end{align*} 我们找到了该恒等式中整数的充要条件，证明了此类恒等式只有有限多个，并给出了生成许多有趣变体的方法。
引用
@article{arxiv.1904.09063,
title = {On a Curious Identity of Ramanujan},
author = {Hung Viet Chu},
journal= {arXiv preprint arXiv:1904.09063},
year = {2020}
}
备注
The author was an undergraduate at Washington and Lee University. The author wants to thank Prof. Abrams Aaron, Kevin Beanland, and Gregory Dresden at Washington and Lee University for many helpful conversations. Special thanks to Prof. Steven Miller at Williams College for valuable comments on the earlier drafts of this paper