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Problems involving arithmetic progressions
Problems involving arithmetic progressions are of interest in number theory, 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. The elements of the forbidden progressions are not required to be consecutive. For example, A4(10) = 8, because {1, 2, 3, 5, 6, 8, 9, 10} has no arithmetic progressions of length 4, while all 9-element subsets of {1, 2, ..., 10} have one.
In 1936, Paul Erdős and Pál Turán posed a question related to this number and Erdős set a $1000 prize for an answer to it. The prize was collected by Endre Szemerédi for a solution published in 1975, what has become known as Szemerédi's theorem.
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.
See also Dirichlet's theorem on arithmetic progressions.
As of 2020[update], the longest known arithmetic progression of primes has length 27:
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Problems involving arithmetic progressions
Problems involving arithmetic progressions are of interest in number theory, 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. The elements of the forbidden progressions are not required to be consecutive. For example, A4(10) = 8, because {1, 2, 3, 5, 6, 8, 9, 10} has no arithmetic progressions of length 4, while all 9-element subsets of {1, 2, ..., 10} have one.
In 1936, Paul Erdős and Pál Turán posed a question related to this number and Erdős set a $1000 prize for an answer to it. The prize was collected by Endre Szemerédi for a solution published in 1975, what has become known as Szemerédi's theorem.
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.
See also Dirichlet's theorem on arithmetic progressions.
As of 2020[update], the longest known arithmetic progression of primes has length 27: