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Ascending and having no a multiple of p

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February 14, 2015
inductionrationumber theoryrelatively primenumber theory proposed

Problem Statement

A sequence {an}n≥1\{a_n\}_{n\geq 1} of positive integers is called ascending if ana_n satisfies an<an+1a_n<a_{n+1} and a2n=2ana_{2n}=2a_n.
(1) Let {an}\{a_n\}be ascending. If pp is a prime greater than a1a_1, then prove that there exists a multiple of pp in the sequence.
(2) Let pp be an odd prime. Prove that there exists a sequence {an}\{a_n\} which is ascending and has no multiple of pp.
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