From Goldbach conjecture to modern arithmetic combinatorics 

Dmitrii Zhelezov received his Ph.D. in Mathematics from Chalmers University of Technology in 2016. Thanks to a grant from the Knut and Alice Wallenberg Foundation, he will hold a postdoctoral position with Professor Endre Szemerédi at Alfréd Rényi Institute of Mathematics in Budapest, Hungary.  

The aim of the project is to study arithmetic combinatorics, a branch of number theory. In contrast to other branches of mathematics, combinatorics focuses on specific problems, which are easy to formulate but notoriously difficult to solve, even when sometimes no mathematical apparatus beyond high school level is necessary for the solution. 

One of the most famous problems in number theory is the Goldbach conjecture, which states that every even integer greater than two can be expressed as a sum of two prime numbers. In 1742, German mathematician Christian Goldbach formulated the conjecture in a letter to a Swiss mathematician, Leonard Euler, by then already recognized as a genius. Euler’s response confirmed that he thought it was true but he couldn’t prove it. Nor has anyone else been able to do so, in the almost 300 years since. The difficulty in proving the conjecture stems from the fact that it refers to addition, whereas prime numbers are defined through multiplication – an integer is prime if it can be divided only by 1 and by itself. However, not much is known about how prime numbers should be linked to addition.  

 A set of sums consists of the sums of all pairs of elements in a given set. The Goldbach conjecture asks whether the set of sums of prime numbers contains all even numbers greater than two. Initially, easy to handle questions about sets of sums can be studied. Subsequently, more advanced new tools and sophisticated methods, which had been developed for other problems, can be applied. 

Sets of sums have many other applications as well. The methods and concepts of abstract number theory have been successfully applied to modern cryptography. They are used to process our credit card payments as well as when we surf the internet. 

Photo: Setta Aspström