Erdős' conjecture on arithmetic progressions, often incorrectly referred to as the Erdős–Turán conjecture, is a conjecture in additive combinatorics due to Paul Erdős. It states that if the sum of the reciprocals of the members of a set A of positive integers diverges, then A contains arbitrarily long arithmetic progressions. Formally, if
- <math> \sum_{n\in\mathbb{A}} \frac{1}{n} =\infty </math>
then A contains arithmetic progressions of any given length. If true, the theorem would generalize Szemerédi's theorem. Erdős offered a prize of $3000 for a proof of this conjecture.[1] The Green-Tao theorem on arithmetic progressions in the primes is a special case of this conjecture.
References
- P. Erdős: Résultats et problèmes en théorie de nombres, Séminaire Delange-Pisot-Poitou (14e année: 1972/1973), Théorie des nombres, Fasc 2., Exp. No. 24, pp. 7,
- P. Erdős: Problems in number theory and combinatorics, Proc. Sixth Manitoba Conf. on Num. Math., Congress Numer. XVIII(1977), 35-58.
- P. Erdős: On the combinatorial problems which I would most like to see solved, Combinatorica, 1(1981), 28.
- ^ Bollobás, Béla (March 1988). "To Prove and Conjecture: Paul Erdős and His Mathematics". American Mathematical Monthly 105 (3): 233.
