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a_n = a_{n-1}^2 + 1 , a_k divides a_{\ell }

Source: Switzerland - 2016 Swiss MO Final Round p6

January 14, 2023
number theorySequencerecurrence relation

Problem Statement

Let ana_n be a sequence of natural numbers defined by a1=ma_1 = m and for n>1n > 1. We call apair(ak,aℓ) (a_k, a_{\ell }) interesting if (i) 0<ℓ−k<20160 < \ell - k < 2016, (ii) aka_k divides aℓa_{\ell }. Show that there exists a mm such that the sequence ana_n contains no interesting pair.
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