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Relative primes in an arithmetic sequence

Source: Kürschak 2007, problem 2

July 8, 2014
arithmetic sequencenumber theoryrelatively primenumber theory unsolved

Problem Statement

Prove that if from any 20072007 consecutive terms of an infinite arithmetic progression of integers starting with 22, one can choose a term relatively prime to all the 20062006 other terms, then there is also a term amongst any 20082008 consecutive terms relatively prime to the rest.