• The French-Peruvian mathematician will talk about patterns and enigmas of prime numbers in the Mathematics at the Residence series on Monday, 30 June at 7:00 p.m.
• In 2013, Helfgott solved Goldbach’s weak conjecture, which had been open since the 18th century, becoming one of the world’s leading experts on the subject.

Harald Helfgott is one of the teachers in Escuela JAE 2025. Image courtesy of the researcher.

Ágata Timón (ICMAT-CSIC)

Few mathematical problems arouse more curiosity than those involving prime numbers. On the one hand, they are an elementary concept that is learned in school and with which everyone is familiar. At the same time, they appear in great enigmas of modern mathematics, such as the Riemann hypothesis, and, moreover, the answers to many elementary questions about them—for example, about their distribution—are unknown. However, for Harald Helfgott, one of the world’s leading experts on these topics, they are normal, everyday numbers: ‘Primes are, from many points of view, very normal numbers; the problem is proving it,’ says the French-Peruvian mathematician, speaker at the next event in the Mathematics at the Residence series: “Enigmas and patterns of prime numbers. From Goldbach to twin primes.” It will take place on Monday 30 June at 7pm at the Residencia de Estudiantes.

‘Both prime numbers and integers seem to follow relatively simple random patterns, but proving that they do is very difficult,’ says Helfgott, principal researcher at the CNRS at the Institut de Mathématiques de Jussieu (University of Paris VI/VII). There are several conjectures, some of them very well known, that fall under this same idea. One of them, which has been popularised through books and films, is Goldbach’s conjecture, which states that every even number greater than two can be written as the sum of two primes. In 2013, Helfgott obtained a proof of the weak version of the conjecture (which states that every odd number can be written as the sum of three primes), a problem that had been open since the 18th century.

Another question, intensely studied by the mathematical community, is the twin prime conjecture: are there an infinite number of “consecutive” primes, such as 5 and 7, 11 and 13, or 2027 and 2029? One more: if a number has an even or odd number of prime factors, does this affect the number of prime factors that the next two numbers will have? ‘These conjectures, far from being mere curiosities, allow us to explore profound questions about integers and their atoms, the primes,’ reflects Helfgott, who is also one of the teachers at the ICMAT’s JAE School of Mathematics, which offers an introduction to research for undergraduate students.

Far from being a strange exception, these statements remain true, in the vast majority of cases, when applied to other infinite sets of numbers. ‘Precisely for this reason, these questions share a profound meaning,’ says Helfgott.

The greatest advance in an impossible problem

The tremendous difficulty involved in proving these conjectures, in contrast to the simplicity of their statements, is undisputed within the mathematical community. Some even claim that new tools, not yet available, need to be developed to prove these statements.

In 2013, Harald Helfgott made the greatest breakthrough to date on Goldbach’s conjecture. He proved its weak version: any odd number greater than 5 can be expressed as the sum of three primes. The first steps towards solving the conjecture were taken by G. H. Hardy and J. E. Littlewood in the early 20th century; later, in the late 1930s, Ivan Vinogradov made great strides. Helfgott used an approach based on improvements to classical tools of analytic number theory, combined with the use of computers.

Helfgott decided to use rigorous computational results to deal with a finite number of objects—small odd numbers, zeros of the Riemann zeta function up to a given height—and focused on improving existing theoretical techniques to such an extent that, in conjunction with finite calculations, they were sufficient to prove the statement in its entirety. He thus managed to close a problem that had been open for 270 years. His proof has been accepted by the Annals of Mathematics Studies series.