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<title>GAP (3k+1) - Chapter 1: The 3k+1 Problem</title>
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<div class="ChapSects"><a href="chap1.html#X8769FB038733F6F4">1 <span class="Heading">The <span class="SimpleMath">3k+1</span> Problem</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1.html#X8729B87B848E3F89">1.1 <span class="Heading">Theory</span></a>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1.html#X7A4605C07A94C9F6">1.2 <span class="Heading">Program</span></a>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X7B1A31B8870EA092">1.2-1 ThreeKPlusOneSequence</a></span>
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<h3>1 <span class="Heading">The <span class="SimpleMath">3k+1</span> Problem</span></h3>
<p><a id="X8729B87B848E3F89" name="X8729B87B848E3F89"></a></p>
<h4>1.1 <span class="Heading">Theory</span></h4>
<p>Let <span class="SimpleMath">k ∈ ℕ</span> be a natural number. We consider the sequence <span class="SimpleMath">n(i, k), i ∈ ℕ,</span> with <span class="SimpleMath">n(1, k) = k</span> and else <span class="SimpleMath">n(i+1, k) = n(i, k) / 2</span> if <span class="SimpleMath">n(i, k)</span> is even and <span class="SimpleMath">n(i+1, k) = 3 n(i, k) + 1</span> if <span class="SimpleMath">n(i, k)</span> is odd.</p>
<p>It is not known whether for any natural number <span class="SimpleMath">k ∈ ℕ</span> there is an <span class="SimpleMath">m ∈ ℕ</span> with <span class="SimpleMath">n(m, k) = 1</span>.</p>
<p><strong class="pkg">ThreeKPlusOne</strong> provides the function <code class="func">ThreeKPlusOneSequence</code> (<a href="chap1.html#X7B1A31B8870EA092"><span class="RefLink">1.2-1</span></a>) to explore this for given <span class="SimpleMath">n</span>. If you really want to know something about this problem, see <a href="chapBib.html#biBWi98">[Wir98]</a> or <span class="URL"><a href="http://www.ku.de/mgf/mathematik/lehrstuhlstatistik/team/dr-guenther-wirsching/">http://www.ku.de/mgf/mathematik/lehrstuhlstatistik/team/dr-guenther-wirsching/</a></span> for more details (and forget this package).</p>
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<h4>1.2 <span class="Heading">Program</span></h4>
<p>In this section we describe the main function of this package.</p>
<p><a id="X7B1A31B8870EA092" name="X7B1A31B8870EA092"></a></p>
<h5>1.2-1 ThreeKPlusOneSequence</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ThreeKPlusOneSequence</code>( <var class="Arg">k</var>[, <var class="Arg">max</var>] )</td><td class="tdright">( function )</td></tr></table></div>
<p>This function computes for a natural number <var class="Arg">k</var> the beginning of the sequence <span class="SimpleMath">n(i, k)</span> defined in section <a href="chap1.html#X8729B87B848E3F89"><span class="RefLink">1.1</span></a>. The sequence stops at the first <span class="SimpleMath">1</span> or at <span class="SimpleMath">n(<var class="Arg">max</var>, k)</span>, if <var class="Arg">max</var> is given.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">ThreeKPlusOneSequence(101);</span>
"Sorry, not yet implemented. Wait for Version 84 of the package"
</pre></div>
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