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Sum of integer powers belongs in A, while their sum doesn't

Source: Romanian IMO TST 2006, day 5, problem 1

May 23, 2006

number theory unsolvednumber theory

Problem Statement

Let

n

be a positive integer of the form

4k+1

k\in \mathbb N

and

A = \{ a^2 + nb^2 \mid a,b \in \mathbb Z\}

. Prove that there exist integers

x,y

such that

x^n+y^n \in A

and

x+y \notin A