Erdős' conjecture on arithmetic progressions, often referred to as the Erdős–Turán conjecture, is a conjecture in arithmetic combinatorics (not to be confused with the Erdős–Turán conjecture on additive bases). 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 arithmetic progressions.

Formally, the conjecture states that if A is a large set in the sense that

$${\displaystyle \sum _{n\in A}{\frac {1}{n}}\ =\ \infty ,}$$

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 ${\displaystyle \{a,a{+}c,a{+}2c,\ldots ,a{+}kc\}\subset A}$.

In 1936, Erdős and Turán made the weaker conjecture that any set of integers with positive natural density contains infinitely many 3 term arithmetic progressions.^[1] This was proven by Klaus Roth in 1952, and generalized to arbitrarily long arithmetic progressions by Szemerédi in 1975 in what is now known as Szemerédi's theorem.

In a 1976 talk titled "To the memory of my lifelong friend and collaborator Paul Turán," Paul Erdős offered a prize of US$3000 for a proof of this conjecture.^[2] As of 2008 the problem is worth US$5000.^[3]

Erdős' conjecture on arithmetic progressions can be viewed as a stronger version of Szemerédi's theorem. Because the sum of the reciprocals of the primes diverges, the Green–Tao theorem on arithmetic progressions is a special case of the conjecture.

The weaker claim that A must contain infinitely many arithmetic progressions of length 3 is a consequence of an improved bound in Roth's theorem. A 2016 paper by Bloom^[4] proved that if ${\displaystyle A\subset \{1,..,N\}}$ contains no non-trivial three-term arithmetic progressions then ${\displaystyle |A|\ll N(\log {\log {N}})/\log {N}}$.

In 2020 a preprint by Bloom and Sisask^[5] improved the bound to ${\displaystyle |A|\ll N/(\log {N})^{1+c}}$ for some absolute constant ${\displaystyle c>0}$ .

In 2023 a new bound of ${\displaystyle \exp(-c(\log N)^{1/12})N}$^[6][7][8] was found by computers scientist Kelley an Meka and shortly after an exposition in more familiar mathematical terms was given by Bloom and Sisask^[9][10] who have since also improved the exponent of the Kelly-Meka bound to ${\displaystyle \beta =1/9}$ (and conjectured ${\displaystyle \beta =5/41}$) in a preprint.^[11]

• Problems involving arithmetic progressions
• List of sums of reciprocals
• List of conjectures by Paul Erdős
• Müntz–Szász theorem

• P. Erdős: Résultats et problèmes en théorie de nombres, Séminaire Delange-Pisot-Poitou (14e année: 1972/1973), Théorie des nombres, Fasc 2., Exp. No. 24, pp. 7,
• P. Erdős and P. Turán, On some sequences of integers, J. London Math. Soc. 11 (1936), 261–264.
• P. Erdős: Problems in number theory and combinatorics, Proc. Sixth Manitoba Conf. on Num. Math., Congress Numer. XVIII(1977), 35–58.
• P. Erdős: On the combinatorial problems which I would most like to see solved, Combinatorica, 1(1981), 28. doi:10.1007/BF02579174

• The Erdős–Turán conjecture or the Erdős conjecture? on MathOverflow