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Find all b's

Source: Polish Math Olympiad 2023 2nd stage P1

February 10, 2023

number theoryDivisibility

Problem Statement

Find all positive integers

b

with the following property: there exists positive integers

a,k,l

such that

a^k + b^l

and

a^l + b^k

are divisible by

b^{k+l}

where

k \neq l