Erdős offered a 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 for an asymptotic formula for , the largest possible cardinality of a subset of that does not contain a 3-term arithmetic progression.

# Erdős-Turán Conjecture

## 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