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If S_k and S_(k+1) are integers, then all S_n are integers

Source: IMO LongList 1973 - P8

June 6, 2011

number theory proposednumber theory

Problem Statement

Let

a

be a non-zero real number. For each integer

n

, we define

S_n = a^n + a^{-n}

. Prove that if for some integer

k

, the sums

S_k

and

S_{k+1}

are integers, then the sums

S_n

are integers for all integers

n