THE legendary mathematician Srinivasa Ramanujan was known聽for coming up with unconventional mathematical ideas. He has now inspired a computer program that does the聽same.
Called the Ramanujan Machine, the software poses conjectures for聽generating equations whose output is fundamental mathematical constants such as 蟺 and e. A聽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聽came to him in a dream,聽and disliked the formal proofs聽favoured by most mathematicians. Ramanujan moved to the UK in 1914 to study at聽the University of Cambridge with the mathematician G.鈥塇. Hardy, and their long friendship led to a series of important results in the field of number theory.
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鈥淩amanujan had a way of producing things which looked true [but] he couldn鈥檛 necessarily convince other people why they were true,鈥� says Saul Schleimer at聽the University of Warwick, UK. Many of Ramanujan鈥檚 conjectures were later formally proven.
The theorems Ramanujan produced often involved continued fractions, which express a number as the sum of聽infinitely nested fractions.
To mimic this approach, Gal Raayoni at the Israel Institute of Technology and his colleagues created . It has already come up with tens of聽conjectures that use continued fractions to approximate 蟺 and e.
One method the program uses to search for new conjectures is a 鈥渕eet in the middle鈥� 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鈥�, 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聽be true. This approach gave the new conjecture shown above.
Schleimer likens the method to聽an extensive process of trial and聽error. 鈥淲hat they鈥檙e doing is聽a聽nice piece of experimental mathematics,鈥� he says. 鈥淏ut it鈥檚 not like this is a new way of thinking.鈥�
Some of the formulas the Ramanujan Machine has come up聽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聽is impossible to prove they are correct with simple arithmetic since they involve infinite sums.
鈥淚t produces conjectures without exactly knowing why they鈥檙e true and it likes continued fractions, which Ramanujan was very, very fond of,鈥夆�� says Schleimer. But it can鈥檛 really match him, he says. 鈥淩amanujan鈥檚 continued fractions were more subtle and in聽some sense more mature.鈥�
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鈥檛 in use. Any conjectures a participant discovers will be named after them, says the team.
