A note on a theorem of I. Niven

Abstract

Niven's proof is based on a theorem of L. J. Mordell [1] on the representability of a binary quadratic form as a sum of two squares of linear forms. The object of this note is to give a simple direct proof of the above theorem. The proof provides explicit formulas for the representations of those Gaussian integers which can be written as a sum of two squares. PROOF. The condition a/2 and b not both integral and odd is necessary. For suppose a =2a'+ 2ib with a' and b odd can be written in the form oz=(x+iy)2+(z+iw)2. This gives the equations