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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 remains open. 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. The conjecture is problem 3 in the Erdős problems collection (Bloom 2026).

See also

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

Portions of this entry contributed by Kevin O'Bryant

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References

Bloom, T. F. "Erdős Problem 3." Erdős Problems. Oct. 1, 2026. https://www.erdosproblems.com/3.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." Ann. Math. 167, 481-547, 2008. https://doi.org/10.4007/annals.2008.167.481.

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Erdős-Turán Conjecture

Cite this as:

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

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