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There exist infinitely many i with gcd(a_i, a_i+1) ≤ 3i/4

Source: China TST 2011 - Quiz 3 - D2 - P2

May 20, 2011
ceiling functionfloor functionnumber theory unsolvednumber theory

Problem Statement

Let a1,a2,…,an,…a_1,a_2,\ldots,a_n,\ldots be any permutation of all positive integers. Prove that there exist infinitely many positive integers ii such that gcd⁡(ai,ai+1)≤34i\gcd(a_i,a_{i+1})\leq \frac{3}{4} i.
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