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Problem 153
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发表于 2007-05-04
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<h1 id="Problem_153"><a href="https://projecteuler.net/problem=153" target="_blank" rel="external">Problem 153</a></h1><hr>
<p><b>Investigating Gaussian Integers</b></p>
<p>As we all know the equation x<sup>2</sup>=-1 has no solutions for real x.<br>If we however introduce the imaginary number i this equation has two solutions: x=i and x=-i.<br>If we go a step further the equation (x-3)<sup>2</sup>=-4 has two complex solutions: x=3+2i and x=3-2i.<br>x=3+2i and x=3-2i are called each others’ complex conjugate.<br>Numbers of the form a+bi are called complex numbers.<br>In general a+bi and a−bi are each other’s complex conjugate.</p>
<p>A Gaussian Integer is a complex number a+bi such that both a and b are integers.<br>The regular integers are also Gaussian integers (with b=0).<br>To distinguish them from Gaussian integers with b ≠ 0 we call such integers “rational integers.”<br>A Gaussian integer is called a divisor of a rational integer n if the result is also a Gaussian integer.<br>If for example we divide 5 by 1+2i we can simplify $\frac{5}{1+2i}$ in the following manner:<br>Multiply numerator and denominator by the complex conjugate of 1+2i: 1−2i.<br>The result is $\frac{5}{1+2i}=\frac{5}{1+2i}\frac{1-2i}{1-2i}=\frac{5(1-2i)}{1-(2i)^2}=\frac{5(1-2i)}{1-(-4)}=\frac{5(1-2i)}{5}=1-2i$.<br>So 1+2i is a divisor of 5.<br>Note that 1+i is not a divisor of 5 because $\frac{5}{1+i}=\frac{5}{2}-\frac{5}{2}i$.<br>Note also that if the Gaussian Integer (a+bi) is a divisor of a rational integer n, then its complex conjugate (a−bi) is also a divisor of n.</p>
<p>In fact, 5 has six divisors such that the real part is positive: {1, 1 + 2i, 1 − 2i, 2 + i, 2 − i, 5}.<br>The following is a table of all of the divisors for the first five positive rational integers:</p>
<table>
<thead>
<tr>
<th style="text-align:center">n</th>
<th style="text-align:center">Gaussian integer divisors with positive real part</th>
<th style="text-align:center">Sum s(n) of these divisors</th>
</tr>
</thead>
<tbody>
<tr>
<td style="text-align:center">1</td>
<td style="text-align:center">1</td>
<td style="text-align:center">1</td>
</tr>
<tr>
<td style="text-align:center">2</td>
<td style="text-align:center">1, 1+i, 1-i, 2</td>
<td style="text-align:center">5</td>
</tr>
<tr>
<td style="text-align:center">3</td>
<td style="text-align:center">1, 3</td>
<td style="text-align:center">4</td>
</tr>
<tr>
<td style="text-align:center">4</td>
<td style="text-align:center">1, 1+i, 1-i, 2, 2+2i, 2-2i,4</td>
<td style="text-align:center">13</td>
</tr>
<tr>
<td style="text-align:center">5</td>
<td style="text-align:center">1, 1+2i, 1-2i, 2+i, 2-i, 5</td>
<td style="text-align:center">12</td>
</tr>
</tbody>
</table>
<p>For divisors with positive real parts, then, we have: $\sum_{n=1}^{5}s(n)=35$.</p>
<p>For 1 ≤ n ≤ 10<sup>5</sup>, ∑ s(n)=17924657155.</p>
<p>What is ∑ s(n) for 1 ≤ n ≤ 10<sup>8</sup>?</p>
<hr>
<p><b>高斯整数的研究</b></p>
<p>我们都知道方程x<sup>2</sup>=-1在实数范围内无解。<br>但如果我们引入虚数i，这个方程将会有两个解x=i和x=-i。<br>进一步地，方程(x-3)<sup>2</sup>=-4有两个复数解：x=3+2i和x=3-2i。<br>x=3+2i和x=3-2i互称为共轭复数。<br>形如a+bi的数被称为复数。<br>概括地说，a+bi和a−bi互称为共轭复数。</p>
<p>高斯整数是形如a+bi且a和b均为整数的复数。<br>一般意义上的整数也是高斯整数（取b=0）。<br>为了把它们和b ≠ 0的高斯整数区分开来，称它们为“有理整数”。<br>如果一个高斯整数除有理整数n的结果仍然是高斯整数，则称它为该有理整数的约数。<br>例如，我们用1+2i除5，按如下方式简化$\frac{5}{1+2i}$：<br>分子和分母同时乘以1+2i的共轭：1−2i。<br>结果是：$\frac{5}{1+2i}=\frac{5}{1+2i}\frac{1-2i}{1-2i}=\frac{5(1-2i)}{1-(2i)^2}=\frac{5(1-2i)}{1-(-4)}=\frac{5(1-2i)}{5}=1-2i$。<br>所以1+2i是5的约数。<br>注意1+i不是5的约数，因为$\frac{5}{1+i}=\frac{5}{2}-\frac{5}{2}i$。<br>同时注意如果高斯整数(a+bi)是有理整数n的约数，那么它的共轭复数(a−bi)也是n的约数。</p>
<p>事实上，5一共有六个实数部分是正数的约数：{1, 1 + 2i, 1 − 2i, 2 + i, 2 − i, 5}。<br>如下表格列出了前五个正有理整数的所有约数：</p>
<table>
<thead>
<tr>
<th style="text-align:center">n</th>
<th style="text-align:center">实数部分是正数的高斯整数约数</th>
<th style="text-align:center">约数的和s(n)</th>
</tr>
</thead>
<tbody>
<tr>
<td style="text-align:center">1</td>
<td style="text-align:center">1</td>
<td style="text-align:center">1</td>
</tr>
<tr>
<td style="text-align:center">2</td>
<td style="text-align:center">1, 1+i, 1-i, 2</td>
<td style="text-align:center">5</td>
</tr>
<tr>
<td style="text-align:center">3</td>
<td style="text-align:center">1, 3</td>
<td style="text-align:center">4</td>
</tr>
<tr>
<td style="text-align:center">4</td>
<td style="text-align:center">1, 1+i, 1-i, 2, 2+2i, 2-2i,4</td>
<td style="text-align:center">13</td>
</tr>
<tr>
<td style="text-align:center">5</td>
<td style="text-align:center">1, 1+2i, 1-2i, 2+i, 2-i, 5</td>
<td style="text-align:center">12</td>
</tr>
</tbody>
</table>
<p>对于实数部分为正数的约数，我们有：$\sum_{n=1}^{5}s(n)=35$。</p>
<p>对于1 ≤ n ≤ 10<sup>5</sup>，∑ s(n)=17924657155。</p>
<p>对于1 ≤ n ≤ 10<sup>8</sup>，求∑ s(n)。</p>
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var oe = event.originalEvent;
var delta = oe.wheelDelta || -oe.detail;
var self = this;
this.scrollTop += ( delta < 0 ? 1 : -1 ) * 30;
event.preventDefault();
toggleTOCOverflowIndicators();
});
}
function tocOverflowIndicator (indicator, action) {
$(indicator).velocity('stop').velocity({
opacity: action === 'show' ? 0.4 : 0
}, { duration: 100 });
}
});
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$(document).ready(function () {
var html = $('html');
$('.sidebar-nav li').on('click', function () {
var item = $(this);
var activeTabClassName = 'sidebar-nav-active';
var activePanelClassName = 'sidebar-panel-active';
if (item.hasClass(activeTabClassName)) {
return;
}
var currentTarget = $('.' + activePanelClassName);
var target = $('.' + item.data('target'));
currentTarget.velocity('transition.slideUpOut', 200, function () {
target
.velocity('stop')
.velocity('transition.slideDownIn', 200)
.addClass(activePanelClassName);
});
item.siblings().removeClass(activeTabClassName);
item.addClass(activeTabClassName);
});
$('.post-toc a').on('click', function (e) {
e.preventDefault();
var offset = $(escapeSelector(this.getAttribute('href'))).offset().top;
html.velocity('stop').velocity('scroll', {
offset: offset + 'px',
mobileHA: false
});
});
// Expand sidebar on post detail page by default, when post has a toc.
var $tocContent = $('.post-toc-content');
if (isDesktop() && CONFIG.sidebar === 'post') {
if ($tocContent.length > 0 && $tocContent.html().trim().length > 0) {
displaySidebar();
}
}
});
</script>
<script type="text/javascript">
$(document).ready(function () {
if (CONFIG.sidebar === 'always') {
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}
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