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for every number c, there exists n, such that a_{n+1} -a_n > c,

Source: Rioplatense 1999 L3 P2

September 19, 2022
inequalitiesnumber theory

Problem Statement

Let p1,p2,...,pkp_1, p_2, ..., p_k be kk different primes. We consider all positive integers that use only these primes (not necessarily all) in their prime factorization, and arrange those numbers in increasing order, forming an infinite sequence: a1<a2<...<an<...a_1 < a_2 < ... < a_n < ... Prove that, for every number cc, there exists nn such that an+1−an>ca_{n+1} -a_n > c.
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