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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="../foundation-test/index.html">foundation-test</a></li>
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<li><a href="index.html"><span class="selectedlink">goldbach</span></a></li>
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<!--Weave of 'Summing Two Primes' generated by 7-->
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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>Summing Two Primes</b></li></ul><p class="purpose">Here we verify the conjecture for small numbers.</p>
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<p class="inwebparagraph"><a id="SP1"></a><b>§1. </b>So, this is a program to see if even numbers from 4 to 100 can all be written
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as a sum of two primes. Christian Goldbach asked Euler in 1742 if every even
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number greater than 2 can be written this way. This remains open, though
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every even number is a sum of at most six primes (Ramaré, 1995) and every
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odd number is a sum of at most five (Tao, 2012).
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</p>
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<center><img src="Letter.jpg"></center>
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<p class="inwebparagraph">Computer verification has been made up to around \(10^{18}\), but by rather better
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methods.
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</p>
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<pre class="definitions">
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<span class="definitionkeyword">define</span> <span class="constant">RANGE</span><span class="plain"> </span><span class="constant">100</span>
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</pre>
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<pre class="display">
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<span class="plain">#</span><span class="identifier">include</span><span class="plain"> <</span><span class="identifier">stdio</span><span class="plain">.</span><span class="identifier">h</span><span class="plain">></span>
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<span class="reserved">int</span><span class="plain"> </span><span class="functiontext">main</span><span class="plain">(</span><span class="reserved">int</span><span class="plain"> </span><span class="identifier">argc</span><span class="plain">, </span><span class="reserved">char</span><span class="plain"> *</span><span class="identifier">argv</span><span class="plain">[]) {</span>
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<span class="reserved">for</span><span class="plain"> (</span><span class="reserved">int</span><span class="plain"> </span><span class="identifier">i</span><span class="plain">=4; </span><span class="identifier">i</span><span class="plain"><</span><span class="constant">RANGE</span><span class="plain">; </span><span class="identifier">i</span><span class="plain">=</span><span class="identifier">i</span><span class="plain">+2)</span>
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<<span class="cwebmacro">Solve Goldbach's conjecture for i</span> <span class="cwebmacronumber">1.1</span>><span class="character">;</span>
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<span class="plain">}</span>
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</pre>
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<p class="inwebparagraph"></p>
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<p class="inwebparagraph"><a id="SP1_1"></a><b>§1.1. </b>This ought to print:
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</p>
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<pre class="display">
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<span class="plain">4 = 2+2</span>
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<span class="plain">6 = 3+3</span>
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<span class="plain">8 = 3+5</span>
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<span class="plain">10 = 3+7 = 5+5</span>
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<span class="plain">12 = 5+7</span>
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<span class="plain">14 = 3+11 = 7+7</span>
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<span class="plain">...</span>
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</pre>
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<p class="inwebparagraph">We'll print each different pair of primes adding up to i. We
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only check in the range \(2 \leq j \leq i/2\) to avoid counting pairs
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twice over (thus \(8 = 3+5 = 5+3\), but that's hardly two different ways).
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</p>
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<p class="macrodefinition"><code class="display">
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<<span class="cwebmacrodefn">Solve Goldbach's conjecture for i</span> <span class="cwebmacronumber">1.1</span>> =
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</code></p>
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<pre class="displaydefn">
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<span class="identifier">printf</span><span class="plain">(</span><span class="string">"%d"</span><span class="plain">, </span><span class="identifier">i</span><span class="plain">);</span>
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<span class="reserved">for</span><span class="plain"> (</span><span class="reserved">int</span><span class="plain"> </span><span class="identifier">j</span><span class="plain">=2; </span><span class="identifier">j</span><span class="plain"><=</span><span class="identifier">i</span><span class="plain">/2; </span><span class="identifier">j</span><span class="plain">++)</span>
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<span class="reserved">if</span><span class="plain"> ((</span><span class="functiontext"><a href="S-tsoe.html#SP2">isprime</a></span><span class="plain">(</span><span class="identifier">j</span><span class="plain">)) && (</span><span class="functiontext"><a href="S-tsoe.html#SP2">isprime</a></span><span class="plain">(</span><span class="identifier">i</span><span class="plain">-</span><span class="identifier">j</span><span class="plain">)))</span>
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<span class="identifier">printf</span><span class="plain">(</span><span class="string">" = %d+%d"</span><span class="plain">, </span><span class="identifier">j</span><span class="plain">, </span><span class="identifier">i</span><span class="plain">-</span><span class="identifier">j</span><span class="plain">);</span>
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<span class="identifier">printf</span><span class="plain">(</span><span class="string">"\n"</span><span class="plain">);</span>
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</pre>
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<p class="inwebparagraph"></p>
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<p class="endnote">This code is used in <a href="#SP1">§1</a>.</p>
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<ul class="toc"><li><i>(This section begins Sections.)</i></li><li><a href="S-tsoe.html">Continue with 'The Sieve of Eratosthenes'</a></li></ul><hr class="tocbar">
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