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Erdős' conjecture on arithmetic progressions, often referred to as the Erdős–Turán conjecture, is a conjecture in arithmetic combinatorics (not to be confused with the Erdős–Turán conjecture on additive bases). 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, the conjecture states that if :: then ''A'' contains arithmetic progressions of any given length. (Sets satisfying the hypothesis are called large sets.) ==History== In 1936, Erdős and Turán made the weaker conjecture that any set of integers with positive natural density contains infinitely many 3 term arithmetic progressions.〔.〕 This was proven by Klaus Roth in 1952, and generalized to arbitrarily long arithmetic progressions by Szemerédi in 1975 in what is now known as Szemerédi's theorem. In a 1976 talk titled "To the memory of my lifelong friend and collaborator Paul Turán," Paul Erdős offered a prize of US$3000 for a proof of this conjecture.〔''Problems in number theory and Combinatorics'', in Proceedings of the Sixth Manitoba Conference on Numerical Mathematics (Univ. Manitoba, Winnipeg, Man., 1976), ''Congress. Numer.'' XVIII, 35–58, Utilitas Math., Winnipeg, Man., 1977〕 The problem is currently worth US$5000.〔p. 354, Soifer, Alexander (2008); The Mathematical Coloring Book: Mathematics of Coloring and the Colorful Life of its Creators; New York: Springer. ISBN 978-0-387-74640-1〕 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Erdős conjecture on arithmetic progressions」の詳細全文を読む スポンサード リンク
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