---
tag: FN978TDX67
kind: conjecture
name: Erdős conjecture on arithmetic progressions
msc: [11B25, 05D10]
uses: [B23B0QEMN5]
about: [B23B0QEMN5]
sources:
  - url: https://en.wikipedia.org/wiki/Erd%C5%91s_conjecture_on_arithmetic_progressions
same_as:
  wikidata: Q1991239
  wikipedia: Erdős conjecture on arithmetic progressions
history:
  posed_by: "Paul Erdős"
---

## Statement

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

## Remarks

The conjecture is open for each $k \ge 4$.
The case $k = 3$ was proved by Bloom and Sisask in 2020, using quantitative bounds in [[Roth's theorem|E7H8NWXESF]].
The set of primes satisfies the hypothesis, since the sum of the reciprocals of the primes diverges.
So the [[Green–Tao theorem|H8BAHWSEMC]] is a special case of the conjecture.
Every set of positive upper density satisfies the hypothesis, so the conjecture would strengthen [[Szemerédi's theorem|B23B0QEMN5]].

## Explanations

### L1 human

The conjecture says that a set of whole numbers that is not too sparse must contain evenly spaced rows of every length.
"Not too sparse" means that the sum of the reciprocals of its elements is infinite.
The primes are an example, and for them the answer is known to be yes.
For general sets of this kind, the question is open beyond rows of three.