Erdős conjecture on arithmetic progressions

Erdős' conjecture on arithmetic progressions, often referred to as the Erdős–Turán conjecture, is a <a href="/facts/Conjecture/oPWUb7SF">conjecture</a> in <a href="/facts/Arithmetic_combinatorics/TXKYrRWs">arithmetic combinatorics</a> (not to be confused with the <a href="/facts/Erd%25C5%2591s%25E2%2580%2593Tur%25C3%25A1n_conjecture_on_additive_bases/MXQHYSxk">Erdős–Turán conjecture on additive bases</a>). It states that if the sum of the reciprocals of the members of a set A of positive integers diverges, then A contains arbitrarily long <a href="/facts/Arithmetic_progression/c4ew9Xoq">arithmetic progressions</a>.
Formally, the conjecture states that if A is a <a href="/facts/Large_set_(combinatorics)/euw8QsX5">large set</a> in the sense that

 
 
 
 
 ∑
 
 n
 ∈
 A
 
 
 
 
 1
 n
 
 
  
 =
  
 ∞
 ,
 
 
 {\displaystyle \sum _{n\in A}{\frac {1}{n}}\ =\ \infty ,}

then A contains arithmetic progressions of any given length, meaning that for every positive integer k there are an integer a and a non-zero integer c such that 
 
 
 
 {
 a
 ,
 a
 
 +
 
 c
 ,
 a
 
 +
 
 2
 c
 ,
 …
 ,
 a
 
 +
 
 k
 c
 }
 ⊂
 A
 
 
 {\displaystyle \{a,a{+}c,a{+}2c,\ldots ,a{+}kc\}\subset A}
 
.

Erdős conjecture on arithmetic progressions open-in-new

Erdős conjecture on arithmetic progressions