The Ramanujan Machine: Using Algorithms for the Discovery of Conjectures on Mathematical Constants

Tuesday, April 6, 2021 - 1:25pm - 2:25pm
Ido Kaminer (Technion-Israel Institute of Technology)

In the past, new conjectures about fundamental constants were discovered sporadically by famous mathematicians such as Newton, Euler, Gauss, and Ramanujan. The talk will present a different approach – a systematic algorithmic approach that discovers new mathematical conjectures on fundamental constants. We call this approach “the Ramanujan Machine”. The algorithms found dozens of well-known formulas as well as previously unknown ones, such as continued fraction representations of π, e, Catalan’s constant, and values of the Riemann zeta function. Part of the conjectures were in retrospect simple to prove, whereas others remained so far unproved. We will discuss these puzzles and wider open questions that arose from this algorithmic investigation – specifically, a newly-discovered algebraic structure that seems to generalize all the known formulas and connect between fundamental constants. We will also discuss two algorithms that proved useful in finding conjectures: a variant of the meet-in-the-middle algorithm and a gradient descent algorithm tailored to the recurrent structure of continued fractions. Both algorithms are based on matching numerical values; consequently, they conjecture formulas without providing proofs or requiring prior knowledge of the underlying mathematical structure. This way, our approach reverses the conventional usage of sequential logic in formal proofs; instead, using numerical data to unveil mathematical structures and provide leads to further mathematical research.

Ido Kaminer joined the Technion as an assistant professor and an Azrieli Faculty Fellow in 2018, after a postdoc at MIT as a Rothschild Fellow, MIT-Technion Fellow, and a Marie Curie Fellow. In his PhD, Ido discovered new classes of accelerating beams in nonlinear optics and electromagnetism, for which he received the 2012 Israel Physical Society Prize, and the 2014 APS (American Physical Society) Award for Outstanding Doctoral Dissertation in Laser Science. Ido was the first Israeli to ever win an APS award for his PhD thesis. He was chosen to the 2020 list of 40 promising leaders under 40 by TheMarker and won multiple awards and grants recently including the ERC Starting Grant, and the 2021 Blavatnik Award for Young Scientists in Israel.