Erdős conjecture on arithmetic progressions

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Erdős' conjecture on arithmetic progressions, often referred to as the Erdős–Turán conjecture, is a conjecture in arithmetic combinatorics. 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 A is a large set in the sense that ∑ n ∈ A 1 n = ∞ , {\displaystyle \sum _{n\in A}{\frac {1}{n}}\ =\ \infty ,} then A contains arithmetic progressions of any given length, meaning that for every positive integer k there are an integer a and a non-zero integer c such that { a , a + c , a + 2...

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