Researchers at the Israel Institute of Technology (Technion) have used artificial intelligence and automation to create a conjecture generator capable of creating these mathematical problems that are the basis of the theorems. The conclusions of what its authors, students of different faculties under the tutelage of assistant professor Ido Kaminer, from the Faculty of Electrical Engineering Andrew; and Erna Viterbi, from the Technion, have dubbed the “Ramanujan machine”, they have just been published in the journal “Nature”.
The project deals with one of the most fundamental elements of mathematics: mathematical constants. A mathematical constant is a number with a fixed value that arises naturally from different mathematical calculations and structures in different fields, although they apply to other fields such as biology, physics, or ecology. For example, the golden ratio and Euler’s number are examples of such fundamental constants. But perhaps the most famous constant is pi, which was studied in ancient times in the context of the circumference of a circle. Today pi appears in numerous formulas in all branches of science, and even many math buffs compete over who can remember the most digits after the decimal point.
WHAT IS A MATHEMATICAL CONJECTURE?
Going back to the study, what the Technion researchers propose is a new idea: use computer algorithms to automatically generate mathematical guesses that appear as formulas for mathematical constants. By parts: a conjecture is a mathematical conclusion or proposition that has not been proved; once the conjecture is proven, it becomes a theorem. In other words, they are the basis of mathematical theories. The discovery of a mathematical conjecture about fundamental constants is relatively rare and often arises from the genius of a mathematician or sudden intuition or inspiration. This was demonstrated by Newton, Riemann, Goldbach, Gauss, Euler, or Ramanujan himself, who gives this machine its name.
Srinivasa Ramanujan, an Indian mathematician born in 1887, grew up in a poor family but managed to reach Cambridge at the age of 26 through the efforts of the British mathematicians Godfrey Hardy and John Littlewood, who saw in him a self-taught genius who had reached conclusions very advanced math with hardly any instruction. Within a few years, he fell ill and returned to India, where he died at the age of 32. However, despite his short life, he achieved great feats in the world of mathematics. One of Ramanujan’s rare capabilities was the intuitive formulation of unproven mathematical formulas, just like this new machine can now do. That is, Ramanujan’s machine “mimics” human intuition (or Ramanujan’s) using artificial intelligence and a high degree of computer automation.
Generating guesses without being able to prove them.
As Kaminer explains: “The computer does not care whether testing the formula is easy or difficult, and does not base the new results on any prior mathematical knowledge, but only on the numbers in the mathematical constants. To a large extent, our algorithms work in the same way as Ramanujan himself, who presented results without evidence. The author also specifies that the algorithm is unable to test the conjectures it found, so “this task is relegated to human mathematicians.
The conjectures generated by this mechanism have produced new formulas for mathematical constants known as pi, the Euler number, the Apéry constant (which is related to the Riemann zeta function), and the Catalan constant. Surprisingly, the algorithms developed by the researchers managed not only to create known formulas for these famous constants but also to uncover various conjectures that were hitherto unknown. The researchers estimate that this algorithm will be able to significantly speed up the generation of mathematical guesses about fundamental constants and help to identify new relationships between these constants. More effective than the “prince of mathematics”
In fact, in hundreds of years of research, only a few dozen formulas have been found. For example, it took Gauss a lifetime to discover the formulas for pi. Ramanujan’s machine took only a few hours, and even discovered dozens more than the one dubbed the “prince of mathematics” had no time to discover. According to the researchers, “similar ideas may in the future lead to the development of mathematical conjectures in all areas of mathematics and thus provide a meaningful tool for mathematical research.
The research team has launched a website whose aim is to inspire the public to participate more in the advancement of mathematical research by providing algorithmic tools that will be available to mathematicians and the general public. Even before the article was published, hundreds of students, experts, and amateur mathematicians had signed up.
And the efforts are already paying off: a high school student named Yahel Manor has managed to discover, thanks to Ramanujan’s machine, a new algebraic structure hidden within a Catalan constant.