In 2025, things are moving for mathematical conjectures!

A priori, 2025 is not a particular number. Square of 45, it admits fifteen divisors, is therefore part of the composite numbers, but it is neither prime nor palindrome nor perfect nor Fibonacci. On the other hand, the year 2025 could be important for mathematics, and, why not, mark the end of decades and sometimes centuries of waiting for great conjectures. Because in this area, there is something new.

Earlier this month, a mathematician at Rutgers University-New Brunswick, Pham Tiep, managed to solve two open problems that date back more than sixty years, including the Brauer conjecture. Which may seem recent in comparison with other conjectures, such as that linked to twin prime numbersopen for more than two millennia and stipulating that there exists an infinity of primes two units apart. One of the problems solved by Tiep was posed by Richard Brauer in 1955 and concerns the theory of finite groups. Simply put, Tiep has uncovered a hidden rule that helps us better understand organization and symmetries in nature and science. But this news may just be a taste of what lies ahead.

The Holy Grail of Math

There is indeed also something new concerning the Holy Grail of mathematics, the Everest of research, namely the inaccessible Riemann hypothesis. So of course, it is neither solved nor proven – if that had been the case, even the television news would have talked about it – but there is nevertheless a breakthrough concerning it. The fame of the Riemann hypothesis lies in its relationship with the distribution of prime numbers. To demonstrate it would be to prove that there is a hidden order behind them. Its formulation, however, is extremely difficult. It is a question of the zeros of a function (that is to say the values ​​where it vanishes) – the Riemann zeta function not to name it – and of the real part of these zeros, supposed always be the same, i.e. ½, stigmatizing their distribution on the same line. The relationship with the first lies in the fact that the famous zeta function (infinite sum) can be matched to a product which is also infinite relating to all the prime numbers.

Infinite and continuous

The novelty is the work of two researchers, James Maynard (who is also the mathematician who determined the smallest gap reproducing infinitely between two consecutive primes) and Larry Guth, who reformulated the entire problem. Assuming that if a zero of the zeta function has a real part distinct from 1/2, then it should be associated with a polynomial of Dirichlet carrying very high value. It therefore remained for them to prove that the said polynomials, in the present case, cannot take on such a large value. Have the two men laid the groundwork for a future demonstration? We want to say yes, especially since their method is reminiscent of that ofAndrew Wileswho demonstrated Fermat’s last theorem in 1994 after more than 350 years using mathematical tools (elliptic curves and modular forms) a priori unrelated to number theory. But things are also shaking up on the side of the hypothesis of the continuum, which has the particularity of being the first problem of the Hilbert’s list and having been proven undecidable in 1963.

Are we sure? No, exactly. This hypothesis, which also requires considerable mathematical baggage to be understood, questions the existence of a set whose cardinal (number which is used to measure the size of sets) would be located between that of the natural integers and that of the real numbers, both infinite. Different demonstrations, including one, famous, of Kurt Gödel in 1938, return the solutions to the hypothesis back to back. Hence its undecidability. In other words, whether it is true or false has no bearing on the set theory on which it depends. According to Georg Cantorfather of transfinite numbers and the theories that result from them, it must be true or false, any other alternative tending to demonstrate that the understanding that we can have on the infinite is artificial. More simply, undecidability does not resolve the question of infinity.

Recently discussed on sites and magazines, the problem is therefore on the table. Will other math puzzles follow? In Hilbert’s list established in 1900 and previously cited, there remain five unresolved, plus a few partially resolved. This is without counting the conjectures outside this list (the existence or not of Lychrel namesthe conjecture de Syracuseetc). All of them have a price, remember. We hope to be able to talk about it again soon.

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