Problems involving arithmetic progressions are of interest in number theory,[1] combinatorics, and computer science, both from theoretical and applied points of view.
Find the cardinality (denoted by Ak(m)) of the largest subset of {1, 2, ..., m} which contains no progression of k distinct terms.
Szemerédi's theorem states that a set of natural numbers of non-zero upper asymptotic density contains finite arithmetic progressions, of any arbitrary length k. Erdős made a more general conjecture from which it would follow that This result was proven by Ben Green and Terence Tao in 2004 and is now known as the Green–Tao theorem.
As of 2020[update], the longest known arithmetic progression of primes has length 27:[4] As of 2011, the longest known arithmetic progression of consecutive primes has length 10.
The prime number theorem for arithmetic progressions deals with the asymptotic distribution of prime numbers in an arithmetic progression.