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Prove the two conditions are equivalent

Source: Turkey IMO TST 1995 #5

July 8, 2011

number theory unsolvednumber theory

Problem Statement

Let

n\in\mathbb{N}

be given. Prove that the following two conditions are equivalent:
(\text{i})\: n|a^n-a for any positive integer

a

; (\text{ii})\: For any prime divisor

p

n

p^2 \nmid n

and

p-1|n-1