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a^{n+1} + b^{n+1} is divisible by a^n + b^n for infinitely n

Source: Tournament of Towns, Senior O-Level , Spring 2019 p2

May 11, 2020

number theorySum of powersDivisibility

Problem Statement

Consider two positive integers

a

and

b

such that

a^{n+1} + b^{n+1}

is divisible by

a^n + b^n

for infinitely many positive integers

n

. Is it necessarily true that

a = b

?
(Boris Frenkin)