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Erdős-Turán Conjecture

Erdős offered a $3000 prize for a proof of the proposition that "If the sum of reciprocals of a set of integers diverges, then that set contains arbitrarily long arithmetic progressions." This conjecture is still open (unsolved), even for 3-term arithmetic progressions. Erdős also offered $10000 for an asymptotic formula for rho_3(n), the largest possible cardinality of a subset of {1,2,...,n} that does not contain a 3-term arithmetic progression.

See also

A-Sequence, B2-Sequence, Szemerédi's Theorem

This entry contributed by Kevin O'Bryant

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References

Erdős, P. and Turán, P. "On Some Sequences of Integers." J. London Math. Soc. 11, 261-264, 1936.Green, B. and Tao, T. "The Primes Contain Arbitrarily Long Arithmetic Progressions." Preprint. 8 Apr 2004. http://arxiv.org/abs/math.NT/0404188.

Cite this as:

O'Bryant, Kevin. "Erdős-Turán Conjecture." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/Erdos-TuranConjecture.html

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