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Number theory: sequence

Source: Iran 3rd round 2017 Number theory first exam-P2

August 9, 2017

number theorynumber theory with sequencesSequence

Problem Statement

Consider a sequence

\{a_i\}^\infty_{i\ge1}

of positive integers. For all positvie integers

n

prove that there exists infinitely many positive integers

k

such that there is no pair

(m,t)

of positive integers where

m>n

and

kn+a_n=tm(m+1)+a_m