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'sums of two squares'
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| Title of object: |
sums of two squares |
| Canonical Name: |
SumsOfTwoSquares |
| Type: |
Theorem |
| Created on: |
2004-05-10 08:56:42 |
| Modified on: |
2005-02-26 03:43:22 |
| Classification: |
msc:11A67 |
Preamble:
% 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 |
Content:
\textbf{Theorem.} \, The set of the sums of two squares of integers is \PMlinkescapetext{closed} under multiplication; in fact we have the identic equation
$$(a^2+b^2)(c^2+d^2) = (ac-bd)^2+(ad+bc)^2.$$
This was proven by Leonardo Fibonacci in 1202.
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. |
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