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China TST 2001 natural number sequence

Source: China TST 2001, problem 2

May 22, 2005
algebra unsolvedalgebracombinatoricsnumber theory

Problem Statement

aa and bb are natural numbers such that b>a>1b > a > 1, and aa does not divide bb. The sequence of natural numbers {bn}n=1∞\{b_n\}_{n=1}^\infty satisfies bn+1≥2bn∀n∈Nb_{n + 1} \geq 2b_n \forall n \in \mathbb{N}. Does there exist a sequence {an}n=1∞\{a_n\}_{n=1}^\infty of natural numbers such that for all n∈Nn \in \mathbb{N}, an+1−an∈{a,b}a_{n + 1} - a_n \in \{a, b\}, and for all m,l∈Nm, l \in \mathbb{N} (mm may be equal to ll), am+al∉{bn}n=1∞a_m + a_l \not\in \{b_n\}_{n=1}^\infty?
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