Researchers have taken things to the next level by developing an AI that is basically a mathematical conjecture generator. Conjectures are mathematical statements that are suspected to be true but have not yet been rigorously proven. Any mathematician will tell you that these are their bread and butter, which they use to develop mathematical theorems. Now, we have computers that can feed mathematicians with new conjectures, which they’ll have to prove, and in the process might revolutionize the field.
The AI developed by the team at Technion-Israel Institute of Technology specifically deals with conjectures surrounding another fundamental element of mathematics: constants. In math, constants are key numbers with fixed values that emerge naturally from different mathematical computations and structures.
Take for instance pi, arguably the most important constant in mathematics. It gives the ratio between a circle’s circumference and diameter, which stays the same value for every circle, no matter how large. Other important fundamental constants include Euler’s number and the golden ratio.
Not anyone can make conjectures about such fundamental constants. In fact, this is something typically reserved for geniuses like Newton, Riemann, Gauss, or Srinivasa Ramanujan. The latter was so good at it that Ramanujan is credited for the discovery of thousands of innovative formulas in number theory — and he did so with no formal training, starting from a poor family background.
In honor of the great mathematician, the researchers named their AI the Ramanujan Machine. Like the late Indian genius, they hope that the AI becomes just as prolific at conjecturing unproven mathematical formulas.
The software has made its own conjectures that independently formulate well-known mathematical constants such as pi, Euler’s number (e), Apéry’s constant, and the Catalan constant, as well as a couple of original universal constants.
“Our results are impressive because the computer doesn’t care if proving the formula is easy or difficult, and doesn’t base the new results on any prior mathematical knowledge, but only on the numbers in mathematical constants. To a large degree, our algorithms work in the same way as Ramanujan himself, who presented results without proof.”
“It’s important to point out that the algorithm itself is incapable of proving the conjectures it found — at this point, the task is left to be resolved by human mathematicians,” said Assistant Professor Ido Kaminer from the Faculty of Electrical Engineering at the Technion.
For thousands of years of mathematical history, conjectures were reserved for rare genius. This is why we only have a few dozen important formulas discovered in the last hundred years of research. But in a few hours, the Ramanujan Machine “re-discovered” all the formulas for pi discovered by Gauss, which took him a lifetime of work, as well as dozens of new formulas that were unknown to Gauss.
“Similar ideas can in the future lead to the development of mathematical conjectures in all areas of mathematics, and in this way provide a meaningful tool for mathematical research,” wrote the researchers in their study published in Nature.
The researchers launched a website where the public can find algorithmic tools that anyone can use for the advancement of mathematical research.
