On a Curious Identity of Ramanujan

TL;DR

This paper investigates a specific identity by Ramanujan, establishing conditions for its validity, proving finiteness of such identities, and offering a method to generate related variations.

Contribution

It provides necessary and sufficient conditions for the identity's integers, proves only finitely many such identities exist, and introduces a method to generate variations.

Findings

Identifies necessary and sufficient conditions for the identity.

Proves the finiteness of such identities.

Provides a method to generate variations.

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.