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< / ul > < h2 > Foundation Module< / h2 > < ul >
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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 >
< p class = "purpose" > Here we verify the conjecture for small numbers.< / p >
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< p class = "commentary firstcommentary" > < a id = "SP1" > < / a > < b > § 1. < / b > On 7 June 1742, Christian Goldbach wrote a letter from Moscow to Leonhard
Euler in Berlin making "eine conjecture hazardiren" that every even number
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
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
several languages in an elegant cursive script, and was much valued as a
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,
the general belief now is that primes are just plentiful enough, and just
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
< / li > < li > (b) every odd number is a sum of at most five (Tao, 2012).
< / li > < / ul >
< 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
because he considered 1 a prime number, as was normal then, and was sometimes
said as late as the early twentieth century.
< 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
of the great age of Enlightenment letter-writing. He and Euler exchanged
scholarly letters for over thirty years, not something Euler would have
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 >
< a href = "#fnref:2" title = "return to text" > ↩ < / a > < / p > < / li > < / ul >
< p class = "commentary firstcommentary" > < a id = "SP2" > < / a > < b > § 2. < / b > Computer verification has been made up to around \(10^{18}\), but by rather
better methods than the one we use here. We will only go up to:
< / 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 = "character-syntax" > ;< / span >
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< span class = "character-syntax" > }< / span >
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< / pre >
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< p class = "commentary firstcommentary" > < a id = "SP2_1" > < / 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 >
< span class = "ConsoleText-plain-syntax" > 10 = 3+7 = 5+5< / span >
< span class = "ConsoleText-plain-syntax" > 12 = 5+7< / span >
< span class = "ConsoleText-plain-syntax" > 14 = 3+11 = 7+7< / span >
< 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
twice over (thus \(8 = 3+5 = 5+3\), but that's hardly two different ways).
< / 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 >
< 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 >
< 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 >
< 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 >
< 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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< ul class = "progressbar" > < li class = "progressprevoff" > ❮ < / li > < li class = "progresscurrent" > sp< / li > < li class = "progresssection" > < a href = "S-tsoe.html" > tsoe< / a > < / li > < li class = "progressnext" > < a href = "S-tsoe.html" > ❯ < / a > < / li > < / ul > < / div >
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