Math Department (Thesis/2009-10)

Sums of squares

Under the direction of: Paul Friedman

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Lots of fun questions to explore: Which integers can be written as a sum of two squares? $n=x^{2}+y^{2}$; as a sum of three squares? $n=x^{2}+y^{2}+z^{2}$; as a sum of four squares? $n=x^{2}+y^{2}+z^{2}+w^{2}$.

A theorem of Lagrange says that every natural number can be written as a sum of (at most) four squares. Can you prove this? (Perhaps you saw a proof of this in Math 235. If so, let's find another proof!) For a given $n$, in how many different ways can this be done? What if we allow the coefficients of the squared terms to vary? For example, while impossible to write every natural number as a sum of three squares, is there any triple $(a, b, c)\ $ such that every natural number can be written as a sum $n=ax^{2}+by^{2}+cz^{2}\ $?

This is a one (or possibly two) term thesis topic. Prerequisite: Math 235 or Math 221 or permission of the instructor.

Math Department web pages Created: 11 May 2009 Last modified: 11 May 2009 00:00:00 Comments to: math@union.edu

Thesis Topics - P. Friedman Thesis Topics for 2009-2010 Thesis Topics - P. Friedman