On a Curious Identity of Ramanujan
Abstract
Ramanujan wrote the following identity \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*} We find necessary and sufficient conditions for the integers in the identity and prove that there are only finitely many such identities, and provide a method to generate many interesting variations.
Keywords
Cite
@article{arxiv.1904.09063,
title = {On a Curious Identity of Ramanujan},
author = {Hung Viet Chu},
journal= {arXiv preprint arXiv:1904.09063},
year = {2020}
}
Comments
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