Erdős conjecture on arithmetic progressions
- Erdős conjecture on arithmetic progressions
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
:
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. [cite journal |last=Bollobás |first=Béla |authorlink=Béla Bollobás |year=1988 |month=March |title=To Prove and Conjecture: Paul Erdős and His Mathematics |journal=American Mathematical Monthly |volume=105 |issue=3 |pages=233]
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.
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