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Does there exist 2011 numbers?

Source: Brazil MO, Problem 4

October 20, 2011
number theory unsolvednumber theory

Problem Statement

Do there exist 20112011 positive integers a1<a2<…<a2011a_1 < a_2 < \ldots < a_{2011} such that gcd⁡(ai,aj)=aj−ai\gcd(a_i,a_j) = a_j - a_i for any ii, jj such that 1≤i<j≤20111 \le i < j \le 2011?
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