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Nitya Nigam Since the advent of modern computing in the 1950s, computers have played an essential role in mathematical research. They're often used to test conjectures, as they can iterate through long lists of numbers extremely quickly to check whether they satisfy the conditions that have been predicted. They are also used to find specific types of numbers, such as Mersenne primes (you can even join the Great Internet Mersenne Prime Search yourself). However, coming up with conjectures has long been the work of mathematicians themselves. A good mathematical conjecture states something profound and useful within the field. It must be interesting enough to prompt investigation, but not so niche as to have very narrow applications. Getting computers to strike this balance is therefore a tall task.
However, researchers at the Technion in Israel have created an automated conjecturing system called the Ramanujan Machine, named after the famous mathematician Srinivasa Ramanujan. The software has already conjectured original formulas for several mathematical constants (published in this Nature article) including Catalan's number and pi. The software works by leveraging the fact that many such constants are equal to continued fractions (fractions where the denominator is the sum of two terms, one of which is a fraction with a denominator that also contains a fraction, and so on until infinity). Continued fractions have been of mathematical interest both for their aesthetics and what they reveal about constants' fundamental properties. The software works by first finding continued fraction expressions that seem to equal universal constants by choosing arbitrary constants and expressions, then computing each side to a certain level of precision and seeing if they approach each other. If they seem to approach each other, they are computed to higher precision to ensure that it is not a coincidence. Since formulas already exist to compute pi and other constants to an arbitrary level of precision, the only thing in the way of making sure the sides match is computing time. So far, the software has conjectured a formula for Catalan's constant that allows for its fastest computation yet. This is an important milestone for computational mathematics, not just because it shows promise for developing faster methods to compute other constants, but because these automated conjectures may be used to reverse-engineer more theorems, further fueling mathematical innovation,
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