THE legendary mathematician
Srinivasa Ramanujan was known for coming up with unconventional mathematical ideas. He has now inspired a computer program that does the same. Called the Ramanujan Machine, the software poses conjectures for generating equations whose output
is fundamental mathematical constants such as π and
e. A conjecture is an unproven mathematical statement. Born in 1887 in what is now
Tamil Nadu in India, Ramanujan was a self-taught mathematician. He often claimed that his results came to him in a dream, and disliked the formal proofs
favoured by most mathematicians. Ramanujan moved to the UK in 1914 to study at the University of Cambridge with the mathematician G. H. Hardy, and their long friendship led to a series of important results in the field of number theory. “Ramanujan had a way of producing things which looked true [but] he couldn’t necessarily convince other people why they were true,” says Saul Schleimer at the University of Warwick, UK.
Many of Ramanujan’s conjectures were later formally proven. The theorems Ramanujan produced often involved continued fractions, which express a number as the sum of infinitely nested fractions.
To mimic this approach, Gal Raayoni at the Israel Institute of Technology and his colleagues created the Ramanujan Machine. It has already come up with tens of conjectures that use continued fractions to approximate π and
e (
arxiv.org/abs/1907.00205). One method the program uses to search for new conjectures is a “meet in the middle” approach. This involves generating many mathematical expressions, computing their value for a limited number of iterations and eliminating the expressions that give inaccurate results. For example, when trying to approximate e, whose value is a decimal that begins 2.718…, any potential conjectures that yield numbers with a value that is too high or too low are eliminated. Conjectures that appear to work are then calculated for more iterations to identify ones likely to be true. This approach gave the new conjecture shown on the left.
Schleimer likens the method to an extensive process of trial and error. “What they’re doing is a nice piece of experimental mathematics,” he says. “But it’s not like this is a new way of thinking.”
Some of the formulas the Ramanujan Machine has come up with are new, while others have previously been discovered by human mathematicians. The team wants people to submit suggested proofs to the new conjectures, as it is impossible to prove they are correct with
“ Ramanujan had a way of producing things which looked true, but he couldn’t always convince others”
Simple arithmetic since they involve infinite sums. “It produces conjectures without exactly knowing why they’re true and it likes continued fractions, which Ramanujan was very, very fond of, ” says Schleimer. But it can’t really match him, he says. “Ramanujan’s continued fractions were more subtle and in some sense more mature.” The researchers behind the Ramanujan Machine have also shared its software, so anyone can download the programme to run on their own computer while it isn’t in use. Any conjectures a participant discovers will be named after them, says the team.
