Number Theory and its Applications

13 Integers as Sums of Squares

“Arithmetic is numbers you squeeze from your head to your hand to your pencil to your paper till you get the answer.”

– Carl Sandburg

13.1 Introduction

Sum of squares of integers is the sum of two squares, three squares, four squares or n squares. In arithmetic operations we have sum of n terms. But there are many techniques for the calculation of sum of squares. In statistics and basic algebra, we have various results using sum of squares. From beginning many famous mathematicians like Euler, Lagrange and Gauss have contributed a lot to develop the concept of sum of squares. The basic problem they have encountered with is that, whether any integer can be expressed as sum of squares or not.

Obviously, we have ...