Rectangles as sums of squares

Rectangles as sums of squares

In this paper we examine generalisations of the following problem posed by Laczkovich: Given an n × m rectangle with n and m integers, it can be written as a disjoint union of squares; what is the smallest number of squares that can be used? He also asked the corresponding higher dimensional analogue. For the two dimensional case Kenyon proved a tight logarithmic bound but left open the higher ...

Representations as Sums of Squares

This is a translation of a paper [5] I wrote in 1971, and may help for Parimala’s course. Evidently completely outdated, but still may be useful. I changed some notation so as to be compatible with the course.

Representing Integers as Sums of Squares

We study in detail the special case of Waring’s problem when the power k = 2. Ultimately, we prove that four is the least number of squares needed to represent any integer. To this end, we prove that some numbers cannot be represented as sums of two squares, some cannot be represented as sums of three, and all can be represented as sums of four. We also show that numbers of a certain form can b...

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...