Erdős conjecture on arithmetic progressions
Erdős' conjecture on arithmetic progressions, often incorrectly referred to as the Erdős–Turán conjecture (see also Erdős–Turán conjecture on additive bases), 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.
(i.e. A is a large set) then A contains arithmetic progressions of any given length.
Even a weaker claim, that A must contain at least one arithmetic progression of length 3, is open.
- Bollobás, Béla (March 1988). "To Prove and Conjecture: Paul Erdős and His Mathematics". American Mathematical Monthly 105 (3): 233. JSTOR 2589077.
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- 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 and P.Turán, On some sequences of integers, J. London Math. Soc. 11 (1936), 261–264.
- 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. doi:10.1007/BF02579174
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