Problem 3

Part of 2001 Mongolian Mathematical Olympiad

Problems(2)

m|a^(n-1)-b^(n-1) and n|a^(m-1)-b^(m-1), infinitude of solns.

Source: Mongolia MO 2001 Grade 10 P3

a,b

be coprime positive integers with

a

a>b

. Show that there exist infinitely many pairs

(m,n)

of coprime positive integers such that

m\mid a^{n-1}-b^{n-1}

n\mid a^{m-1}-b^{m-1}

number theoryDivisibility

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

Source: Mongolia MO 2001 Teachers P3

k\ge0

be a given integer. Suppose there exists positive integer

n,d

and an odd integer

m>1

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

m\mid n^d+1

. Find all possible values of

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