Sums of Squares: An Elementary Method

Abstract

If x ₁, x ₂, ... , x _s are integers positive negative or zero such that

$$ x_1^2 + x_2^2 + \cdot \cdot \cdot x_s^2 = n, $$

then (x ₁, x ₂, ... , x _s ) is called a representation of n as a sum of s squares, and the total number of representations is denoted by R _s (n). Two representations (x ₁, x ₂, ... , x _s ) and (y ₁, y ₂, ... , y _s ) are considered to be different unless

$$ {x_1} = {y_1},{x_2} = {y_2},...,{x_s} = {y_s}. $$