MathDB

Given positive odd integers

m

and

n

where the set of all prime factors of

m

is the same as the set of all prime factors

n

, and

n \vert m

. Let

a

be an arbitrary integer which is relatively prime to

m

and

n

. Prove that:

o_m(a) = o_n(a) \times \frac{m}{\gcd(m, a^{o_n(a)}-1)}

where

o_k(a)

denotes the smallest positive integer such that

a^{o_k(a)} \equiv 1

(mod

k

) holds for some natural number

k > 1

Problem Statement