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<!--Weave of 'Summing Primes' 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>Summing Primes</b></li></ul></div>
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<p class="purpose">Here we verify the conjecture for small numbers.</p>
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<p class="commentary firstcommentary"><a id="SP1" class="paragraph-anchor"></a><b>§1. </b>On 7 June 1742, Christian Goldbach wrote a letter from Moscow to Leonhard
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Euler in Berlin making "eine conjecture hazardiren" that every even number
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greater than \(2\) can be written as a sum of two primes.<sup id="fnref:1"><a href="#fn:1" rel="footnote">1</a></sup> Euler did not
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know if this was true, and nor does anyone else.
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</p>
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<p class="center-p"><img src="Letter.jpg" alt="Letter.jpg" height="720"></p>
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<p class="commentary">Goldbach, a professor at St Petersburg and tutor to Tsar Peter II, wrote in
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several languages in an elegant cursive script, and was much valued as a
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letter-writer, though his reputation stands less high today.<sup id="fnref:2"><a href="#fn:2" rel="footnote">2</a></sup> All the same,
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the general belief now is that primes are just plentiful enough, and just
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evenly-enough spread, for Goldbach to be right. It is known that:
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</p>
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<ul class="items"><li>(a) every even number is a sum of at most six primes (Ramaré, 1995), and
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</li><li>(b) every odd number is a sum of at most five (Tao, 2012).
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</li></ul>
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<ul class="footnotetexts"><li class="footnote" id="fn:1"><p class="inwebfootnote"><sup id="fnref:1"><a href="#fn:1" rel="footnote">1</a></sup> "Greater than 2" is our later proviso: Goldbach needed no such exception
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because he considered 1 a prime number, as was normal then, and was sometimes
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said as late as the early twentieth century.
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<a href="#fnref:1" title="return to text"> ↩</a></p></li><li class="footnote" id="fn:2"><p class="inwebfootnote"><sup id="fnref:2"><a href="#fn:2" rel="footnote">2</a></sup> Goldbach, almost exactly a contemporary of Voltaire, was a good citizen
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of the great age of Enlightenment letter-writing. He and Euler exchanged
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scholarly letters for over thirty years, not something Euler would have
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kept up with a duffer. Goldbach was also not, as is sometimes said, a lawyer.
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See: <a href="http://mathshistory.st-andrews.ac.uk/Biographies/Goldbach.html" class="external">http://mathshistory.st-andrews.ac.uk/Biographies/Goldbach.html</a>.
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An edited transcription of the letter is at: <a href="http://eulerarchive.maa.org//correspondence/letters/OO0765.pdf" class="external">http://eulerarchive.maa.org//correspondence/letters/OO0765.pdf</a>
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<a href="#fnref:2" title="return to text"> ↩</a></p></li></ul>
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<p class="commentary firstcommentary"><a id="SP2" class="paragraph-anchor"></a><b>§2. </b>Computer verification has been made up to around \(10^{18}\), but by rather
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better methods than the one we use here. We will only go up to:
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</p>
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<pre class="definitions code-font"><span class="definition-keyword">define</span> <span class="constant-syntax">RANGE</span><span class="plain-syntax"> </span><span class="constant-syntax">100</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="plain-syntax">#</span><span class="identifier-syntax">include</span><span class="plain-syntax"> <</span><span class="identifier-syntax">stdio</span><span class="plain-syntax">.</span><span class="identifier-syntax">h</span><span class="plain-syntax">></span>
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<span class="reserved-syntax">int</span><span class="plain-syntax"> </span><span class="function-syntax">main</span><span class="plain-syntax">(</span><span class="reserved-syntax">int</span><span class="plain-syntax"> </span><span class="identifier-syntax">argc</span><span class="plain-syntax">, </span><span class="reserved-syntax">char</span><span class="plain-syntax"> *</span><span class="identifier-syntax">argv</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">i</span><span class="plain-syntax">=4; </span><span class="identifier-syntax">i</span><span class="plain-syntax"><</span><span class="constant-syntax">RANGE</span><span class="plain-syntax">; </span><span class="identifier-syntax">i</span><span class="plain-syntax">=</span><span class="identifier-syntax">i</span><span class="plain-syntax">+2) </span><span class="comment-syntax"> stepping in twos to stay even</span>
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<span class="plain-syntax"> </span><span class="named-paragraph-container code-font"><a href="S-sp.html#SP2_1" class="named-paragraph-link"><span class="named-paragraph">Solve Goldbach's conjecture for i</span><span class="named-paragraph-number">2.1</span></a></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" class="paragraph-anchor"></a><b>§2.1. </b>This ought to print:
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</p>
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<pre class="ConsoleText-displayed-code all-displayed-code code-font">
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<span class="ConsoleText-plain-syntax"> </span><span class="ConsoleText-element-syntax">$</span><span class="ConsoleText-plain-syntax"> </span><span class="ConsoleText-function-syntax">goldbach/Tangled/goldbach</span>
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<span class="ConsoleText-plain-syntax"> 4 = 2+2</span>
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<span class="ConsoleText-plain-syntax"> 6 = 3+3</span>
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<span class="ConsoleText-plain-syntax"> 8 = 3+5</span>
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<span class="ConsoleText-plain-syntax"> 10 = 3+7 = 5+5</span>
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<span class="ConsoleText-plain-syntax"> 12 = 5+7</span>
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<span class="ConsoleText-plain-syntax"> 14 = 3+11 = 7+7</span>
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<span class="ConsoleText-plain-syntax"> ...</span>
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</pre>
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<p class="commentary">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="commentary"><span class="named-paragraph-container code-font"><span class="named-paragraph-defn">Solve Goldbach's conjecture for i</span><span class="named-paragraph-number">2.1</span></span><span class="ConsoleText-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">printf</span><span class="plain-syntax">(</span><span class="string-syntax">"%d"</span><span class="plain-syntax">, </span><span class="identifier-syntax">i</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">j</span><span class="plain-syntax">=2; </span><span class="identifier-syntax">j</span><span class="plain-syntax"><=</span><span class="identifier-syntax">i</span><span class="plain-syntax">/2; </span><span class="identifier-syntax">j</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><a href="S-tsoe.html#SP2" class="function-link"><span class="function-syntax">isprime</span></a><span class="plain-syntax">(</span><span class="identifier-syntax">j</span><span class="plain-syntax">)) && (</span><a href="S-tsoe.html#SP2" class="function-link"><span class="function-syntax">isprime</span></a><span class="plain-syntax">(</span><span class="identifier-syntax">i</span><span class="plain-syntax">-</span><span class="identifier-syntax">j</span><span class="plain-syntax">)))</span>
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<span class="plain-syntax"> </span><span class="identifier-syntax">printf</span><span class="plain-syntax">(</span><span class="string-syntax">" = %d+%d"</span><span class="plain-syntax">, </span><span class="identifier-syntax">j</span><span class="plain-syntax">, </span><span class="identifier-syntax">i</span><span class="plain-syntax">-</span><span class="identifier-syntax">j</span><span class="plain-syntax">);</span>
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<span class="plain-syntax"> </span><span class="identifier-syntax">printf</span><span class="plain-syntax">(</span><span class="string-syntax">"\n"</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-sp.html#SP2">§2</a>.</li></ul>
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