Sums of three and four integer squares

Sums of Two and Four Squares

This document gives the formal proofs of the following results about the sums of two and four squares: 1. Any prime number p ≡ 1 mod 4 can be written as the sum of two squares. 2. (Lagrange) Any natural number can be written as the sum of four

On Sums of Three Squares

(1) r3(n) = 4πn S3(n), where the singular series S3(n) is given by (16) with Q = ∞. While in principle this exact formula can be used to answer almost any question concerning r3(n), the ensuing calculations can be tricky because of the slow convergence of the singular series S3(n). Thus, one often sidesteps (1) and attacks problems involving r3(n) directly. For example, concerning the mean valu...

An asymptotic bound on integer sums of squares

Let σZ(k) be the smallest n such that there exists an identity (x1 + x 2 2 + · · ·+ xk) · (y 1 + y 2 + · · ·+ y k) = f 1 + f 2 + · · ·+ f n , with f1, . . . fn being polynomials with integer coefficients in the variables x1, . . . , xk and y1, . . . , yk. We prove that σZ(k) ≥ Ω(k).

Sums of distinct squares

Sums-of-Squares Formulas

The following is the extended version of my notes from my ATC talk given on June 4, 2014 at UCLA. I begin with a basic introduction to sums-of-squares formulas, and move on to giving motivation for studying these formulas and discussing some results about them over the reals. More recent techniques have made it possible to obtain similar results over arbitrary fields, and some of these are disc...