6

Part of 2016 Switzerland - Final Round

Problems(1)

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

Source: Switzerland - 2016 Swiss MO Final Round p6

a_n

be a sequence of natural numbers defined by

a_1 = m

n > 1

. We call apair

(a_k, a_{\ell })

interesting if (i)

0 < \ell - k < 2016

a_k

a_{\ell }

. Show that there exists a

m

such that the sequence

a_n

contains no interesting pair.

number theorySequencerecurrence relation