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m|n^d+1, find (m^(2^k)-1)/d if integer

Source: Mongolia MO 2001 Teachers P3

April 12, 2021

number theory

Problem Statement

Let

k\ge0

be a given integer. Suppose there exists positive integer

n,d

and an odd integer

m>1

with

d\mid m^{2^k}-1

and

m\mid n^d+1

. Find all possible values of

\frac{m^{2^k}-1}d