10 June, 2021 - 18:58 (Research Matters)

Do you recollect knowing axioms and proving theorems in your high-school mathematics class? Most theorems start as conjectures — a proposition that is believed to be true but without enough formal proof. Over time, mathematicians use the axioms to prove the conjectures. Proving or disproving conjectures can be challenging, sometimes taking centuries. ‘Fermat’s Last Theorem’, which states that no three positive integers can satisfy the equation an+bn=cn where n>2, although sounds straightforward, took 350 years to prove!

How do mathematicians arrive at conjectures in the first place, before even proceeding to prove? While some are easy to guess, the conjectures that appear simple sometimes rely on the mathematician’s intuitive brilliance, which only few individuals seem to have. Proving conjectures, which result in new theorems, provide novel insights between previously known numbers, thus driving forward the field of mathematics. They reveal deeper relationships between fundamental constants that frequently occur in nature and govern how the world around us works, like the pi (π), the exponential (e), or the ‘golden ratio’.

Now, scientists have developed computer algorithms that can spew such conjectures rapidly by looking at patterns of numbers. Proving them, however, is still up to us. This machine is named after the Indian mathematician Srinivasa Ramanujan, who formulated many conjectures, including the one that has his name. The ‘Ramanujan Machine’, as it is called, reproduces known mathematical formulae between fundamental constants of nature and uses them to discover new conjectures on its own. The algorithms powering this machine are a series of calculations carried out in tandem that crunch a series of numbers. In turn, they are formulating unknown relationships between fundamental constants of nature!

So far, the Ramanujan Machine has discovered conjectures involving the ‘Apery’s constant’, the ‘Catalan constant’, and formulae for log(2). Do you have it in you to prove any of these? Head to http://www.ramanujanmachine.com/ and see what are the challenges ahead.