The Ramanujan Machine: Researchers have developed a 'conjecture generator' that creates mathematical conjectures

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Using AI and computer automation, Technion researchers have developed a 'conjecture generator' that creates mathematical conjectures, which are considered to be the starting point for developing mathematical theorems. They have already used it to generate a number of previously unknown formulas. The study, which was published in the journal Nature, was carried out by undergraduates from different faculties under the tutelage of Assistant Professor Ido Kaminer of the Andrew and Erna Viterbi Faculty of Electrical Engineering at the Technion.

 

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 emerges naturally from different mathematical calculations and mathematical structures in different fields. Many mathematical constants are of great importance in mathematics, but also in disciplines that are external to mathematics, including biology, physics, and ecology. The golden ratio and Euler's number are examples of such fundamental constants. 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, with many math aficionados competing over who can recall more digits after the decimal point: 3.1415926535897932384626433832795028841971693993751058209749445923078164062862089986280348253421170 67982148086513282306647093844609550582231725 3594081284811174502841027019385211055596446229489549303820...

 

The Technion researchers proposed and examined a new idea: The use of computer algorithms to automatically generate mathematical conjectures that appear in the form of formulas for mathematical constants.

 

A conjecture is a mathematical conclusion or proposition that has not been proved; once the conjecture is proved, it becomes a theorem. Discovery of a mathematical conjecture on fundamental constants is relatively rare, and its source often lies in mathematical genius and exceptional human intuition. Newton, Riemann, Goldbach, Gauss, Euler, and Ramanujan are examples of such genius, and the new approach presented in the paper is named after Srinivasa Ramanujan.

 

Ramanujan, an Indian mathematician born in 1887, grew up in a poor family, yet managed to arrive in Cambridge at the age of 26 at the initiative of British mathematicians Godfrey Hardy and John Littlewood. Within a few years he fell ill and returned to India, where he died at the age of 32. During his brief life he accomplished great achievements in the world of mathematics. One of Ramanujan's rare capabilities was the intuitive formulation of unproven mathematical formulas. The Technion research team therefore decided to name their algorithm "the Ramanujan Machine," as it generates conjectures without proving them, by "imitating" intuition using AI and considerable computer automation.

 

According to Prof. Kaminer, "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."

 

The conjectures generated by the Technion's Ramanujan Machine have delivered new formulas for well-known mathematical constants such as pi, Euler's number (e), Apéry's constant (which is related to the Riemann zeta function), and the Catalan constant. Surprisingly, the algorithms developed by the Technion researchers succeeded not only in creating known formulas for these famous constants, but in discovering several conjectures that were heretofore unknown.

 

The researchers estimate this algorithm will be able to significantly expedite the generation of mathematical conjectures on fundamental constants and help to identify new relationships between these constants.

 

As mentioned, until now, these conjectures were based on rare genius. This is why in hundreds of years of research, only a few dozens of formulas were found. It took the Technion's Ramanujan Machine just a few hours to discover all the formulas for pi discovered by Gauss, the "Prince of Mathematics," during a lifetime of work, along with dozens of new formulas that were unknown to Gauss.

 

According to the researchers, "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."

 

The research team has launched a website, RamanujanMachine.com, which is intended to inspire the public to be more involved in the advancement of mathematical research by providing algorithmic tools that will be available to mathematicians and the public at large. Even before the article was published, hundreds of students, experts, and amateur mathematicians had signed up to the website.

 

The research study started out as an undergraduate project in the Rothschild Scholars Technion Program for Excellence with the participation of Gal Raayoni and George Pisha, and continued as part of the research projects conducted in the Andrew and Erna Viterbi Faculty of Electrical Engineering with the participation of Shahar Gottlieb, Yoav Harris, and Doron Haviv. This is also where the most significant breakthrough was made—by an algorithm developed by Shahar Gottlieb—which led to the article's publication in Nature. Prof. Kaminer adds that the most interesting mathematical discovery made by the Ramanujan Machine's algorithms to date relates to a new algebraic structure concealed within a Catalan constant.

 

The structure was discovered by high school student Yahel Manor, who participated in the project as part of the Alpha Program for science-oriented youth. Prof. Kaminer added that, "Industry colleagues Uri Mendlovic and Yaron Hadad also participated in the study, and contributed greatly to the mathematical and algorithmic concepts that form the foundation for the Ramanujan Machine. It is important to emphasize that the entire project was executed on a voluntary basis, received no funding, and participants joined the team out of pure scientific curiosity."

 

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AI maths whiz creates tough new problems for humans to solve

 

Algorithm named after mathematician Srinivasa Ramanujan suggests interesting formulae, some of which are difficult to prove true.

 

Srinivasa Ramanujan made important contributions to mathematics in the early twentieth century.Credit: Historic Collection/Alamy

 

Researchers have built an artificial intelligence (AI) that can generate new mathematical formulae — including some as-yet unsolved problems that continue to challenge mathematicians.

 

The Ramanujan Machine is designed to generate new ways of calculating the digits of important mathematical constants, such as π or e, many of which are irrational, meaning they have an infinite number of non-repeating decimals.

 

The AI starts with well-known formulae to calculate the digits — the first few thousand digits of π, for example. From those, the algorithm tries to predict a new formula that does the same calculation just as well. The process produces a good guess called a conjecture — it is then up to human mathematicians to prove that the formula can correctly calculate the whole number.

 

The team began to make the conjectures public on the project’s website in 2019, and researchers have since proved several of them correct. But some remain open questions, including one on Apery’s constant, a number that has important applications in physics. “The last result, the most exciting one, no one knows how to prove,” says physicist Ido Kaminer, who leads the project at the Technion — Israel Institute of Technology in Haifa. The automated creation of conjectures could point mathematicians towards connections between branches of maths that people did not know existed, he adds.

 

The project — described1 in Nature on 3 February — is named after Srinivasa Ramanujan, an Indian mathematician who was active in the early twentieth century. Ramanujan rarely wrote the types of proof that appear in conventional maths papers. Instead, he filled entire notebooks with formulae that he believed came from a goddess who appeared in his dreams. His work has continued to inspire new research long after he died, aged 32, in 1920.

 

The techniques in the Ramanujan Machine’s algorithms existed before, says mathematician Doron Zeilberger at Rutgers University in Piscataway, New Jersey. “The novelty is to combine them in a unified framework.”

 

Continued fractions

The Ramanujan Machine currently has limited applications: so far, the algorithms can generate only formulae of a particular type, called continued fractions. These express a number as an infinite sequence of fractions nested in each other’s denominators.

 

Kaminer’s team has experimented with a range of algorithms for finding continued fractions, and applied them to various conceptually important numbers. One of them is Catalan’s constant, a number that originated from nineteenth-century Belgian mathematician Eugène Catalan’s studies.

 

Catalan’s constant is approximately 0.916, but it is so mysterious that no one has yet worked out whether it is rational — that is, whether it can be expressed as a fraction of two whole numbers. The best mathematicians have been able to do is prove that its ‘irrationality exponent’ — a measure of how hard it is to approximate a number using rationals — is at least 0.554. Proving that Catalan’s constant is irrational would be equivalent to proving that its irrationality exponent is greater than 1. Formulae generated by the Ramanujan Machine have enabled Kaminer’s team to improve slightly on the best human result, bringing the exponent up to 0.567.

 

“The fact that they have improved the irrationality exponent for the Catalan constant from 0.554 to 0.567 reveals that they are able to make contributions to really hard problems,” says George Andrews, a mathematician who has helped to curate the posthumous publication of some of Ramanujan’s notebooks. But the contributions made so far are not of the calibre that using Ramanujan’s name would suggest, says Andrews.

 

“Calling this the Ramanujan Machine is over the top,” says Andrews, who is at Pennsylvania State University in University Park.

 

Kaminer’s team plans to broaden the AI’s technique so that it can generate other kinds of mathematical formula.

 

Increasing complexity

Automated generation of conjectures is not the only place where computers are helping to advance maths. Although many mathematicians prefer working with pencil and paper, standard research practice in the field now includes the use of mathematical software that can, for example, manipulate complicated algebraic expressions.

 

Computer-aided calculations have played a crucial part in producing the proofs of several high-profile results.

 

And more recently, some mathematicians have made progress towards AI that doesn’t just perform repetitive calculations, but develops its own proofs. Another growing area has been software that can go over a mathematical proof written by humans and check that it is correct.

 

“Eventually, humans will be obsolete,” says Zeilberger, who has pioneered automation in proofs and has helped confirm some of the Ramanujan Machine's conjectures2. And as the complexity of AI-generated mathematics grows, mathematicians will lose track of what computers are doing and will be able to understand the calculations only in broad outline, he adds.

 

Andrews says that although computers might be able to come up with mathematical statements, and even prove that they are true, without human intervention, it is unclear whether they will be able to distinguish profound, interesting statements from merely technically correct ones. “Until I can detect a well-developed ‘sense of mathematical taste’ in AI, I expect its role to be that of an important auxiliary tool, not that of independent discoverer.”

 

doi: https://doi.org/10.1038/d41586-021-00304-8

 

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