91 lines
6.5 KiB
HTML
91 lines
6.5 KiB
HTML
<!DOCTYPE html>
|
|
<html lang="en">
|
|
<head>
|
|
<meta charset="utf-8" />
|
|
<meta name="author" content="Colin Hughes" />
|
|
<meta name="description" content="A website dedicated to the fascinating world of mathematics and programming" />
|
|
<meta name="keywords" content="programming,mathematics,problems,puzzles" />
|
|
|
|
<title>Problem 229 - Project Euler</title>
|
|
<link rel="shortcut icon" href="http://projecteuler.net/favicon.ico" />
|
|
<link rel="stylesheet" type="text/css" href="style_main.css" />
|
|
<link rel="stylesheet" type="text/css" href="style_light.css" />
|
|
<script type="text/x-mathjax-config">
|
|
MathJax.Hub.Config({
|
|
jax: ["input/TeX", "output/HTML-CSS"],
|
|
tex2jax: {
|
|
inlineMath: [ ["$","$"], ["\\(","\\)"] ],
|
|
displayMath: [ ["$$","$$"], ["\\[","\\]"] ],
|
|
processEscapes: true
|
|
},
|
|
"HTML-CSS": { availableFonts: ["TeX"] }
|
|
});
|
|
</script>
|
|
|
|
<script type="text/javascript" src="http://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS_HTML">
|
|
</script>
|
|
</head>
|
|
|
|
<body>
|
|
<div id="container">
|
|
|
|
<div id="nav" class="noprint">
|
|
<ul>
|
|
<li><a href="about" title="About" accesskey="h">About</a></li>
|
|
<li><a href="register" title="Register" accesskey="1">Register</a></li>
|
|
<li id="current"><a href="problems" title="Problems" accesskey="2">Problems</a></li>
|
|
<li><a href="login" title="Login" accesskey="3">Login</a></li>
|
|
</ul>
|
|
</div>
|
|
<div id="info_panel"><a href="rss2_euler.xml"><img src="images/icon_rss.png" alt="RSS Feed" title="RSS Feed" /></a><a href="secure=fd7de"><img src="images/icon_lock.png" alt="Use secure connection" title="Use secure connection" /></a></div>
|
|
|
|
<div id="logo" class="noprint">
|
|
<img src="images/pe_banner_light.png" alt="Project Euler .net" />
|
|
</div>
|
|
|
|
<div id="content">
|
|
<div style="text-align:center;" class="print"><img src="images/pe_banner.png" alt="projecteuler.net" style="border:none;" /></div>
|
|
<h2>Four Representations using Squares</h2><div class="info" style="cursor:help;width:200px;margin-bottom:10px;"><h3>Problem 229</h3><span style="width:300px;color:#666;">Published on Saturday, 24th January 2009, 09:00 am; Solved by 826</span></div>
|
|
<div class="problem_content" role="problem">
|
|
<p>Consider the number 3600. It is very special, because</p>
|
|
<div style="margin-left:200px;">
|
|
3600 = 48<sup>2</sup> + 36<sup>2</sup><br /><br />
|
|
3600 = 20<sup>2</sup> + 2<img src='images/symbol_times.gif' width='9' height='9' alt='×' border='0' style='vertical-align:middle;' />40<sup>2</sup><br /><br />
|
|
3600 = 30<sup>2</sup> + 3<img src='images/symbol_times.gif' width='9' height='9' alt='×' border='0' style='vertical-align:middle;' />30<sup>2</sup><br /><br />
|
|
3600 = 45<sup>2</sup> + 7<img src='images/symbol_times.gif' width='9' height='9' alt='×' border='0' style='vertical-align:middle;' />15<sup>2</sup><br /><br />
|
|
</div>
|
|
|
|
<p>Similarly, we find that 88201 = 99<sup>2</sup> + 280<sup>2</sup> = 287<sup>2</sup> + 2<img src='images/symbol_times.gif' width='9' height='9' alt='×' border='0' style='vertical-align:middle;' />54<sup>2</sup> = 283<sup>2</sup> + 3<img src='images/symbol_times.gif' width='9' height='9' alt='×' border='0' style='vertical-align:middle;' />52<sup>2</sup> = 197<sup>2</sup> + 7<img src='images/symbol_times.gif' width='9' height='9' alt='×' border='0' style='vertical-align:middle;' />84<sup>2</sup>.</p>
|
|
|
|
<p>In 1747, Euler proved which numbers are representable as a sum of two squares.
|
|
We are interested in the numbers <var>n</var> which admit representations of all of the following four types:</p>
|
|
<div style="margin-left:200px;">
|
|
<var>n</var> = <var>a<sub>1</sub></var><sup>2</sup> + <var>b<sub>1</sub></var><sup>2</sup><br /><br />
|
|
<var>n</var> = <var>a<sub>2</sub></var><sup>2</sup> + 2 <var>b<sub>2</sub></var><sup>2</sup><br /><br />
|
|
<var>n</var> = <var>a<sub>3</sub></var><sup>2</sup> + 3 <var>b<sub>3</sub></var><sup>2</sup><br /><br />
|
|
<var>n</var> = <var>a<sub>7</sub></var><sup>2</sup> + 7 <var>b<sub>7</sub></var><sup>2</sup>,
|
|
</div>
|
|
<p>where the <var>a</var><sub><var>k</var></sub> and <var>b</var><sub><var>k</var></sub> are positive integers.</p>
|
|
|
|
<p>There are 75373 such numbers that do not exceed 10<sup>7</sup>.<br />
|
|
|
|
How many such numbers are there that do not exceed 2<img src='images/symbol_times.gif' width='9' height='9' alt='×' border='0' style='vertical-align:middle;' />10<sup>9</sup>?</p>
|
|
</div><br />
|
|
<br /></div>
|
|
|
|
|
|
|
|
|
|
|
|
<div id="footer" class="noprint">
|
|
<a href="copyright">Project Euler Copyright Information</a>
|
|
<!--/Creative Commons License--><!-- <rdf:RDF xmlns="http://web.resource.org/cc/" xmlns:dc="http://purl.org/dc/elements/1.1/" xmlns:rdf="http://www.w3.org/1999/02/22-rdf-syntax-ns#" xmlns:rdfs="http://www.w3.org/2000/01/rdf-schema#">
|
|
<Work rdf:about="">
|
|
<license rdf:resource="http://creativecommons.org/licenses/by-nc-sa/2.0/uk/" />
|
|
<dc:type rdf:resource="http://purl.org/dc/dcmitype/Text" />
|
|
</Work>
|
|
<License rdf:about="http://creativecommons.org/licenses/by-nc-sa/2.0/uk/"><permits rdf:resource="http://web.resource.org/cc/Reproduction"/><permits rdf:resource="http://web.resource.org/cc/Distribution"/><requires rdf:resource="http://web.resource.org/cc/Notice"/><requires rdf:resource="http://web.resource.org/cc/Attribution"/><prohibits rdf:resource="http://web.resource.org/cc/CommercialUse"/><permits rdf:resource="http://web.resource.org/cc/DerivativeWorks"/><requires rdf:resource="http://web.resource.org/cc/ShareAlike"/></License></rdf:RDF> -->
|
|
</div>
|
|
</div>
|
|
<div style="height:1px;"> </div></body>
|
|
</html> |