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Golden Residue

Source: Iran MO 3rd round 2016 mid-terms - Number Theory P3

September 6, 2016

number theorymodular arithmeticIran

Problem Statement

Let

m

be a positive integer. The positive integer

a

is called a golden residue modulo

m

if

\gcd(a,m)=1

and

x^x \equiv a \pmod m

has a solution for

x

. Given a positive integer

n

, suppose that

a

is a golden residue modulo

n^n

. Show that

a

is also a golden residue modulo

n^{n^n}

.
Proposed by Mahyar Sefidgaran

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