SUMS OF FOUR SQUARES WITH A CERTAIN RESTRICTION

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 certain permutation groups and sums of two squares

We consider the question of existence of ramified covers over P1 matching certain prescribed ramification conditions. This problem has already been faced in a number of papers, but we discuss alternative approaches for an existence proof, involving elliptic curves and universal ramified covers with signature. We also relate the geometric problem with finite permutation groups and with the Ferma...

Sums of Squares in Certain Quaternion and Octonion Algebras

Formulae for the levels and sublevels of certain quaternion and octonion algebras are established. Corollaries concerning the equality of levels and sublevels of quaternion algebras with those of associated octonion algebras are presented. Let R be a not necessarily associative ring with unity. The level and sublevel of R, respectively denoted by s(R) and s(R), are defined as follows: s(R) = in...

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...