  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.
