Two theorems on square numbers

Abstract

We show that if a is a positive integer such that for each positive integer n, a+n^2 can be expressed x^2+y^2, where x,y\in \mathbb{Z}, then a is a square number. A similar theorem also holds if a+n^2 and x^2+y^2 are replaced by a+2n^2 and x^2+2y^2, respectively.