Math 2.0

Number theory Conjecture

Erdős conjecture on arithmetic progressions

Builds on Szemerédi's theorem

Statement

Let AA be a set of positive integers such that the series ∑a∈A1/a\sum_{a \in A} 1/a diverges. Then for every integer k≥1k \ge 1, the set AA contains an arithmetic progression of length kk with nonzero common difference.

Remarks

The conjecture is open for each k≥4k \ge 4. The case k=3k = 3 was proved by Bloom and Sisask in 2020, using quantitative bounds in Roth's theorem. The set of primes satisfies the hypothesis, since the sum of the reciprocals of the primes diverges. So the Green–Tao theorem is a special case of the conjecture. Every set of positive upper density satisfies the hypothesis, so the conjecture would strengthen Szemerédi's theorem.

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  1. Limit superior concept
  2. Szemerédi's theorem statement
  3. Erdős conjecture on arithmetic progressions conjecture

Further reading

  1. Web https://en.wikipedia.org/wiki/Erd%C5%91s_conjecture_on_arithmetic_progressions
About this page
TagFN978TDX67
URIm2://FN978TDX67
Kindconjecture
WritingW0 stub
MSC11B25, 05D10
StatusThis copy of the site has no build data. Run m2 status FN978TDX67.
AboutSzemerédi's theorem
UsesSzemerédi's theorem
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Changed2 November 2026

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