sums of two squares SumsOfTwoSquares 2004-05-10 08:56:42 2008-05-18 11:41:12 Theorem integer % 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 \theoremstyle{definition} \newtheorem*{thmplain}{Theorem} \begin{thmplain} \,\, The set of the sums of two squares of integers is closed under multiplication; in fact we have the identical equation $$(a^2+b^2)(c^2+d^2) = (ac-bd)^2+(ad+bc)^2.$$ \end{thmplain} This was presented by Leonardo Fibonacci in 1225 (in ``Liber quadratorum''), but the original inventor was Brahmagupta. The proof of the equation may utilize imaginary numbers as follows: \begin{align*} (a^2+b^2)(c^2+d^2) & = (a+ib)(a-ib)(c+id)(c-id)\\ & = (a+ib)(c+id)(a-ib)(c-id)\\ & = [(ac-bd)+i(ad+bc)][(ac-bd)-i(ad+bc)]\\ & = (ac-bd)^2+(ad+bc)^2 \end{align*} \textbf{Note.}\, The equation is the special case\, $n = 2$\, of Lagrange's identity.