Ramanujan evaluated two other continued fractions in this way. It was largely because of those that Hardy, the professional, and Ramanujan, the amateur genius, ended up collaborating to invent some of the most important mathematics of their day.
How did Ramanujan come up with those identities that captivated Hardy?
Ramanujan had a secret device: two more identities that he had discovered, which later came to be called the Rogers-Ramanujan identities. He plugged specific numbers into those two identities.
What did you discover about the Rogers-Ramanujan identities?
Griffin, Warnaar and I found a framework that shows why they’re true and structurally what makes them tick. They turned out to be two golden nuggets that suggested the existence of a whole mother lode of identities out there. And we showed that they can easily produce numbers mathematicians call “algebraic,” which are rare, beautiful numbers akin to the golden ratio.
Why is it important?
The Rogers-Ramanujan identities are tied to a lot of deep mathematics. So this larger framework for them will open up new mathematical territory.
[This article originally appeared in print as "A Beautiful Find."]