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Numbers Theory

Source: Iran 3rd round 2017 Numbers theory final exam-P1

August 30, 2017

number theoryIran 3rd Round

Problem Statement

Let

x

and

y

be integers and let

p

be a prime number. Suppose that there exist realatively prime positive integers

m

and

n

such that

x^m \equiv y^n \pmod p

Prove that there exists an unique integer

z

modulo

p

such that x \equiv z^n \pmod p \text{and} y \equiv z^m \pmod p