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 is an element of 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.