Thue's lemma ThuesLemma2 2002-12-25 15:56:34 2005-03-18 22:31:37 Theorem prime number sum of squares % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here Let $p$ be a prime number of the form $4k+1$ . Then there are two unique integers $a$ and $b$ with $0<a<b$ such that $p=a^2+b^2$. Additionally, if a number $p$ can be written in as the sum of two squares in 2 different ways (i.e. $p=a^2+b^2$ and $p=c^2+d^2$ with the two sums being different), then the number $p$ is composite.