Szemerédi’s theorem

mathematics

Written by

The Information Architects of Encyclopaedia Britannica

Last updated

July 20, 2026

Quick Summary

Szemerédi's theorem, a significant contribution to number theory, addresses arithmetic progressions within sets of natural numbers. An arithmetic progression is a sequence where the difference between consecutive terms is constant, such as 2, 4, 6, 8.1

The theorem, proven by mathematician Endre Szemerédi, states that for any given density of numbers in a set, there will always be an arithmetic progression of any length within that set. This resolved a conjecture made by Paul Erdős and Paul Turán in 1936. Klaus Roth had previously proven the conjecture for three-term progressions, but Szemerédi extended it to progressions of any length. This theorem has also found applications in computer science.1

Summary created from Britannica articles using AI. AI can make mistakes, so verify using Britannica articles.

2 Britannica Sources

Ask Britannica AI about this topic

Britannica doesn't have a full article on this topic, so we've curated a list of our relevant coverage.

Endre Szemerédi
…theorem, which became known as Szemerédi’s theorem, proved a 1936 conjecture by Erdős and Hungarian mathematician Paul Turán. In number theory, an arithmetic progression is a sequence of numbers that proceeds in steps of the same amount. For example, 2, 4, 6, 8 is a progression with four terms and…
Read More
…theorem, which became known as Szemerédi’s theorem, proved a 1936 conjecture by Erdős and Hungarian mathematician Paul Turán. In number theory, an arithmetic progression is a sequence of numbers that proceeds in steps of the same amount. For example, 2, 4, 6, 8 is a progression with four terms and…
Read More