A new artificially intelligent “mathematician” known as the Ramanujan Machine can potentially reveal hidden relationships between numbers.
The “machine” consists of algorithms that seek out conjectures, or mathematical conclusions that are likely true but have not been proved. Conjectures are the starting points of mathematical theorems, which are conclusions that have been proved by a series of equations.
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The set of algorithms is named after Indian mathematician Srinivasa Ramanujan. Born in 1887 to a store clerk and a homemaker, Ramanujan was a child prodigy who came up with many mathematical conjectures, proofs and solutions to equations that had never before been solved. In 1918, two years before his early death from disease, he was elected as a Fellow of The Royal Society London, becoming only the second Indian man to be inducted after marine engineer Ardaseer Cursetjee in 1841.
Ramanujan had an innate feel for numbers and an eye for patterns that eluded other people, said physicist Yaron Hadad, vice president of AI and data science at the medical device company Medtronic and one of the developers of the new Ramanujan Machine. The new AI mathematician is designed to pull out promising mathematical patterns from large sets of potential equations, Hadad told Live Science, making Ramanujan a fitting namesake.
Math by machine
Machine learning, in which an algorithm detects patterns in large amounts of data with minimal direction from programmers, has been put to use in a variety of pattern-finding applications, from image recognition to drug discovery. Hadad and his colleagues at the Technion-Israel Institute of Technology in Haifa wanted to see if they could use machine learning for something more fundamental.
“We wanted to see if we could apply machine learning to something that is very, very basic, so we thought numbers and number theory are very, very basic,” Hadad told Live Science. (Number theory is the study of integers, or numbers that can be written without fractions.)
Already, some researchers have used machine learning to turn conjectures into theorems — a process called automated theorem proving. The goal of the Ramanujan Machine, instead, is to identify promising conjectures in the first place. This has previously been the domain of human mathematicians, who have come up with famous proposals such as Fermat’s Last Theorem, which claims that there are no three positive integers that can solve the equation an + bn = cn when n is greater than 2. (That famous conjecture was scribbled in the margins of a book by mathematician Pierre de Fermat in 1637 but wasn’t proven until 1994.)
To direct the Ramanujan Machine, the researchers focused on fundamental constants, which are numbers that are fixed and fundamentally true across equations. The most famous constant might be the ratio of a circle’s circumference to its diameter, better known as pi. Regardless of the size of the circle, that ratio is always 3.14159265… and on and on.
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The algorithms essentially scan large numbers of potential equations in search of patterns that might indicate the existence of formulas to express such a constant. The programs first scan a limited number of digits, perhaps five or 10, and then record any matches and expand upon those to see if the patterns repeat further.
When a promising pattern appears, the conjecture is then available for an attempt at a proof. More than 100 intriguing conjectures have been generated so far, Hadad said, and several dozen have been proved.
The researchers reported their results Feb. 3 in the journal Nature. They have also set up a website, RamanujanMachine.com, to share the conjectures the algorithms generate and to collect attempted proofs from anyone who’d like to take a stab at discovering a new theorem. Users can also download the code to run their own searches for conjectures, or let the machine use their spare processing space on their own computers to look on its own. Part of the goal, Hadad said, is to get lay people more involved in the world of mathematics.
The researchers also hope that the Ramanujan Machine will help change how math is done. It’s hard to say how advances in number theory will translate to real-world applications, Hadad said, but so far, the algorithm has helped uncover a better measure of irrationality for Catalan’s constant, a number denoted by G that has at least 600,000 digits but may or may not be an irrational number. (An irrational number cannot be written as a fraction; a rational number can.) The algorithm hasn’t yet answered the question of whether Catalan’s constant is or isn’t rational, but it’s moved a step closer to that goal, Hadad said.
“We are still in the very early stages of this project, where the full potential is only starting to unfold,” he told Live Science in an email. “I believe that generalizing this concept to other areas of mathematics and physics (or even other fields of science) will enable researchers to get leads to new research from computers. So human scientists will be able to choose better goals to work on from a wider selection offered by computers, and thus improve their productivity and potential impact on human knowledge and future generations.”
Originally published on Live Science.