Scientists from Technion — Israel Institute of Technology have developed a concept they have named the Ramanujan Machine, after the Indian mathematician. It is not really a machine but an algorithm, and performs a very unconventional function.
What it does
With most computer programs, humans input a problem and expect the algorithm to work out a solution. With the Ramanujan Machine, it works the other way round. Feed in a constant, say the well-know pi, and the algorithm will come up with a equation involving an infinite series whose value, it will propose, is exactly pi. Over to humans now: let someone prove that this proposed equation is correct.
The algorithm reflects the way Srinivasa Ramanujan worked during his brief life (1887-1920). With very little formal training, he engaged with the most celebrated mathematicians of the time, particularly during his stay in England (1914-19), where he eventually became a Fellow of the Royal Society and earned a research degree from Cambridge.
Throughout his life, Ramanujan came up with novel equations and identities —including equations leading to the value of pi — and it was usually left to formally trained mathematicians to prove these. In 1987, two Canadian brothers proved all 17 of Ramanujan’s series for 1/pi; two years earlier, an American mathematician and programmer had used one of these formulas to calculate pi up to over 17 million digits, which was a world record at the time (Deka Baruah, Berndt & Chan; American Mathematical Monthly, 2009).
What’s the point?
Conjectures are a major step in the process of making new discoveries in any branch of science, particularly mathematics. Equations defining the fundamental mathematical constants, including pi, are invariably elegant. New conjectures in mathematics, however, have been scarce and sporadic, the researchers note in their paper, which is currently on a pre-print server. The idea is to enhance and accelerate the process of discovery.
How good is it?
The paper gives examples for previously unknown equations produced by the algorithm, including for values of the constants pi and e. The Ramanujan Machine proposed these conjecture formulas by matching numerical values, without providing proofs. It has to be remembered, however, that these are infinite series, and a human can only enter a finite number of terms to test the value of the series. The question is, therefore, whether the series will fail after a point. The researchers feel this is unlikely, because they tested hundreds of digits.
Until proven, it remains a conjecture. By the same token, until proven wrong, a conjecture remains one. It is quite possible that the algorithm will come up with conjectures that may take years to prove — a famous example of a human conjecture is Fermat’s Last Theorem, proposed in 1637 and proved only in 1994.
Where to find it
The researchers have set up a website, ramanujanmachine.com. Users can suggest proofs for algorithms or propose new algorithms, which will be named after them.