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Set: If n >n_0^2 then n in A

Source: ISI(BS) 2007 #10

April 10, 2012
number theory unsolvednumber theory

Problem Statement

Let AA be a set of positive integers satisfying the following properties: (i) if mm and nn belong to AA, then m+nm+n belong to AA; (ii) there is no prime number that divides all elements of AA.
(a) Suppose n1n_1 and n2n_2 are two integers belonging to AA such that n2−n1>1n_2-n_1 >1. Show that you can find two integers m1m_1 and m2m_2 in AA such that 0<m2−m1<n2−n10< m_2-m_1 < n_2-n_1 (b) Hence show that there are two consecutive integers belonging to AA. (c) Let n0n_0 and n0+1n_0+1 be two consecutive integers belonging to AA. Show that if n≥n02n\geq n_0^2 then nn belongs to AA.
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