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<h1><a href="../index.html">
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<ul><li><a href="../inweb/index.html">inweb</a></li>
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</ul><h2>Foundation Module</h2><ul>
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<li><a href="index.html"><span class="selectedlink">goldbach</span></a></li>
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<!--Weave of 'The Sieve of Eratosthenes' generated by Inweb-->
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<ul class="crumbs"><li><a href="../index.html">Home</a></li><li><a href="index.html">The Goldbach Conjecture</a></li><li><b>The Sieve of Eratosthenes</b></li></ul><p class="purpose">A fairly fast way to determine if small numbers are prime, given storage.</p>
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<ul class="toc"><li><a href="S-tsoe.html#SP1">§1. Storage</a></li><li><a href="S-tsoe.html#SP2">§2. Primality</a></li></ul><hr class="tocbar">
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<p class="commentary firstcommentary"><a id="SP1"></a><b>§1. Storage. </b>This technique, still essentially the best sieve for finding prime
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numbers, is attributed to Eratosthenes of Cyrene and dates from the 200s BC.
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Since composite numbers are exactly those numbers which are multiples of
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something, the idea is to remove everything which is a multiple: whatever
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is left, must be prime.
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</p>
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<p class="commentary">This is very fast (and can be done more quickly than the implementation
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below), but (a) uses storage to hold the sieve, and (b) has to start right
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back at 2 - so it can't efficiently test just, say, the eight-digit numbers
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for primality.
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</p>
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<pre class="displayed-code all-displayed-code code-font">
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<span class="reserved-syntax">int</span><span class="plain-syntax"> </span><span class="identifier-syntax">still_in_sieve</span><span class="plain-syntax">[</span><span class="constant-syntax">RANGE</span><span class="plain-syntax"> + </span><span class="constant-syntax">1</span><span class="plain-syntax">];</span>
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<span class="reserved-syntax">int</span><span class="plain-syntax"> </span><span class="identifier-syntax">sieve_performed</span><span class="plain-syntax"> = </span><span class="constant-syntax">FALSE</span><span class="plain-syntax">;</span>
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</pre>
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<p class="commentary firstcommentary"><a id="SP2"></a><b>§2. Primality. </b>We provide this as a function which determines whether a number is prime:
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</p>
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<pre class="definitions code-font"><span class="definition-keyword">define</span> <span class="constant-syntax">TRUE</span><span class="plain-syntax"> </span><span class="constant-syntax">1</span>
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<span class="definition-keyword">define</span> <span class="constant-syntax">FALSE</span><span class="plain-syntax"> </span><span class="constant-syntax">0</span>
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</pre>
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<pre class="displayed-code all-displayed-code code-font">
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<span class="reserved-syntax">int</span><span class="plain-syntax"> </span><span class="function-syntax">isprime</span><button class="popup" onclick="togglePopup('usagePopup1')"><span class="comment-syntax">?</span><span class="popuptext" id="usagePopup1">Usage of <span class="code-font"><span class="function-syntax">isprime</span></span>:<br>Summing Primes - <a href="S-sp.html#SP2_1">§2.1</a></span></button><span class="plain-syntax">(</span><span class="reserved-syntax">int</span><span class="plain-syntax"> </span><span class="identifier-syntax">n</span><span class="plain-syntax">) {</span>
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<span class="plain-syntax"> </span><span class="reserved-syntax">if</span><span class="plain-syntax"> (</span><span class="identifier-syntax">n</span><span class="plain-syntax"> <= </span><span class="constant-syntax">1</span><span class="plain-syntax">) </span><span class="reserved-syntax">return</span><span class="plain-syntax"> </span><span class="constant-syntax">FALSE</span><span class="plain-syntax">;</span>
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<span class="plain-syntax"> </span><span class="reserved-syntax">if</span><span class="plain-syntax"> (</span><span class="identifier-syntax">n</span><span class="plain-syntax"> > </span><span class="constant-syntax">RANGE</span><span class="plain-syntax">) { </span><span class="identifier-syntax">printf</span><span class="plain-syntax">(</span><span class="string-syntax">"Out of range!\n"</span><span class="plain-syntax">); </span><span class="reserved-syntax">return</span><span class="plain-syntax"> </span><span class="constant-syntax">FALSE</span><span class="plain-syntax">; }</span>
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<span class="plain-syntax"> </span><span class="reserved-syntax">if</span><span class="plain-syntax"> (!</span><span class="identifier-syntax">sieve_performed</span><span class="plain-syntax">) </span><span class="named-paragraph-container code-font"><a href="S-tsoe.html#SP2_1" class="named-paragraph-link"><span class="named-paragraph">Perform the sieve</span><span class="named-paragraph-number">2.1</span></span></a><span class="plain-syntax">;</span>
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<span class="plain-syntax"> </span><span class="reserved-syntax">return</span><span class="plain-syntax"> </span><span class="identifier-syntax">still_in_sieve</span><span class="plain-syntax">[</span><span class="identifier-syntax">n</span><span class="plain-syntax">];</span>
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<span class="plain-syntax">}</span>
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</pre>
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<p class="commentary firstcommentary"><a id="SP2_1"></a><b>§2.1. </b>We save a little time by noting that if a number up to <span class="extract"><span class="extract-syntax">RANGE</span></span> is composite
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then one of its factors must be smaller than the square root of <span class="extract"><span class="extract-syntax">RANGE</span></span>. Thus,
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in a sieve of size 10000, one only needs to remove multiples of 2 up to 100,
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for example.
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</p>
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<p class="commentary"><span class="named-paragraph-container code-font"><span class="named-paragraph-defn">Perform the sieve</span><span class="named-paragraph-number">2.1</span></span><span class="comment-syntax"> =</span>
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</p>
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<pre class="displayed-code all-displayed-code code-font">
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<span class="plain-syntax"> </span><span class="named-paragraph-container code-font"><a href="S-tsoe.html#SP2_1_1" class="named-paragraph-link"><span class="named-paragraph">Start with all numbers from 2 upwards in the sieve</span><span class="named-paragraph-number">2.1.1</span></span></a><span class="plain-syntax">;</span>
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<span class="plain-syntax"> </span><span class="reserved-syntax">for</span><span class="plain-syntax"> (</span><span class="reserved-syntax">int</span><span class="plain-syntax"> </span><span class="identifier-syntax">n</span><span class="plain-syntax">=2; </span><span class="identifier-syntax">n</span><span class="plain-syntax">*</span><span class="identifier-syntax">n</span><span class="plain-syntax"> <= </span><span class="constant-syntax">RANGE</span><span class="plain-syntax">; </span><span class="identifier-syntax">n</span><span class="plain-syntax">++)</span>
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<span class="plain-syntax"> </span><span class="reserved-syntax">if</span><span class="plain-syntax"> (</span><span class="identifier-syntax">still_in_sieve</span><span class="plain-syntax">[</span><span class="identifier-syntax">n</span><span class="plain-syntax">])</span>
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<span class="plain-syntax"> </span><span class="named-paragraph-container code-font"><a href="S-tsoe.html#SP2_1_2" class="named-paragraph-link"><span class="named-paragraph">Shake out multiples of n</span><span class="named-paragraph-number">2.1.2</span></span></a><span class="plain-syntax">;</span>
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<span class="plain-syntax"> </span><span class="identifier-syntax">sieve_performed</span><span class="plain-syntax"> = </span><span class="constant-syntax">TRUE</span><span class="plain-syntax">;</span>
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</pre>
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<ul class="endnotetexts"><li>This code is used in <a href="S-tsoe.html#SP2">§2</a>.</li></ul>
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<p class="commentary firstcommentary"><a id="SP2_1_1"></a><b>§2.1.1. </b><span class="named-paragraph-container code-font"><span class="named-paragraph-defn">Start with all numbers from 2 upwards in the sieve</span><span class="named-paragraph-number">2.1.1</span></span><span class="comment-syntax"> =</span>
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</p>
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<pre class="displayed-code all-displayed-code code-font">
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<span class="plain-syntax"> </span><span class="identifier-syntax">still_in_sieve</span><span class="plain-syntax">[1] = </span><span class="constant-syntax">FALSE</span><span class="plain-syntax">;</span>
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<span class="plain-syntax"> </span><span class="reserved-syntax">for</span><span class="plain-syntax"> (</span><span class="reserved-syntax">int</span><span class="plain-syntax"> </span><span class="identifier-syntax">n</span><span class="plain-syntax">=2; </span><span class="identifier-syntax">n</span><span class="plain-syntax"> <= </span><span class="constant-syntax">RANGE</span><span class="plain-syntax">; </span><span class="identifier-syntax">n</span><span class="plain-syntax">++) </span><span class="identifier-syntax">still_in_sieve</span><span class="plain-syntax">[</span><span class="identifier-syntax">n</span><span class="plain-syntax">] = </span><span class="constant-syntax">TRUE</span><span class="plain-syntax">;</span>
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</pre>
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<ul class="endnotetexts"><li>This code is used in <a href="S-tsoe.html#SP2_1">§2.1</a>.</li></ul>
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<p class="commentary firstcommentary"><a id="SP2_1_2"></a><b>§2.1.2. </b><span class="named-paragraph-container code-font"><span class="named-paragraph-defn">Shake out multiples of n</span><span class="named-paragraph-number">2.1.2</span></span><span class="comment-syntax"> =</span>
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</p>
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<pre class="displayed-code all-displayed-code code-font">
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<span class="plain-syntax"> </span><span class="reserved-syntax">for</span><span class="plain-syntax"> (</span><span class="reserved-syntax">int</span><span class="plain-syntax"> </span><span class="identifier-syntax">m</span><span class="plain-syntax">= </span><span class="identifier-syntax">n</span><span class="plain-syntax">+</span><span class="identifier-syntax">n</span><span class="plain-syntax">; </span><span class="identifier-syntax">m</span><span class="plain-syntax"> <= </span><span class="constant-syntax">RANGE</span><span class="plain-syntax">; </span><span class="identifier-syntax">m</span><span class="plain-syntax"> += </span><span class="identifier-syntax">n</span><span class="plain-syntax">) </span><span class="identifier-syntax">still_in_sieve</span><span class="plain-syntax">[</span><span class="identifier-syntax">m</span><span class="plain-syntax">] = </span><span class="constant-syntax">FALSE</span><span class="plain-syntax">;</span>
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</pre>
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<ul class="endnotetexts"><li>This code is used in <a href="S-tsoe.html#SP2_1">§2.1</a>.</li></ul>
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<ul class="toc"><li><a href="S-sp.html">Back to 'Summing Primes'</a></li><li><i>(This section ends Sections.)</i></li></ul><hr class="tocbar">
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