96 lines
6.4 KiB
HTML
96 lines
6.4 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 140 - 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=6f9e9"><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>Modified Fibonacci golden nuggets</h2><div class="info" style="cursor:help;width:200px;margin-bottom:10px;"><h3>Problem 140</h3><span style="width:300px;color:#666;">Published on Saturday, 3rd February 2007, 07:00 am; Solved by 2035</span></div>
|
|
<div class="problem_content" role="problem">
|
|
<p>Consider the infinite polynomial series A<sub>G</sub>(<i>x</i>) = <i>x</i>G<sub>1</sub> + <i>x</i><sup>2</sup>G<sub>2</sub> + <i>x</i><sup>3</sup>G<sub>3</sub> + ..., where G<sub><i>k</i></sub> is the <i>k</i>th term of the second order recurrence relation G<sub><i>k</i></sub> = G<sub><i>k</i><img src='images/symbol_minus.gif' width='9' height='3' alt='−' border='0' style='vertical-align:middle;' />1</sub> + G<sub><i>k</i><img src='images/symbol_minus.gif' width='9' height='3' alt='−' border='0' style='vertical-align:middle;' />2</sub>, G<sub>1</sub> = 1 and G<sub>2</sub> = 4; that is, 1, 4, 5, 9, 14, 23, ... .</p>
|
|
<p>For this problem we shall be concerned with values of <i>x</i> for which A<sub>G</sub>(<i>x</i>) is a positive integer.</p>
|
|
<p>The corresponding values of <i>x</i> for the first five natural numbers are shown below.</p>
|
|
<div style="text-align:center;">
|
|
<table cellspacing="0" cellpadding="2" border="1" align="center">
|
|
<tr style="background-color:#c1daf9;">
|
|
<td><b><i>x</i></b></td><td width='50'><b>A<sub>G</sub>(<i>x</i>)</b></td>
|
|
</tr>
|
|
<tr>
|
|
<td>(<img src='images/symbol_radic.gif' width='14' height='16' alt='√' border='0' style='vertical-align:middle;' />5<img src='images/symbol_minus.gif' width='9' height='3' alt='−' border='0' style='vertical-align:middle;' />1)/4</td><td>1</td>
|
|
</tr>
|
|
<tr>
|
|
<td>2/5</td><td>2</td>
|
|
</tr>
|
|
<tr>
|
|
<td>(<img src='images/symbol_radic.gif' width='14' height='16' alt='√' border='0' style='vertical-align:middle;' />22<img src='images/symbol_minus.gif' width='9' height='3' alt='−' border='0' style='vertical-align:middle;' />2)/6</td><td>3</td>
|
|
</tr>
|
|
<tr>
|
|
<td>(<img src='images/symbol_radic.gif' width='14' height='16' alt='√' border='0' style='vertical-align:middle;' />137<img src='images/symbol_minus.gif' width='9' height='3' alt='−' border='0' style='vertical-align:middle;' />5)/14</td><td>4</td>
|
|
</tr>
|
|
<tr>
|
|
<td>1/2</td><td>5</td>
|
|
</tr>
|
|
</table>
|
|
</div>
|
|
<p>We shall call A<sub>G</sub>(<i>x</i>) a golden nugget if <i>x</i> is rational, because they become increasingly rarer; for example, the 20th golden nugget is 211345365.</p>
|
|
<p>Find the sum of the first thirty golden nuggets.</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> |