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<!-- Weave of 'Summing Two 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 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
number greater than 2 can be written this way. This remains open, though
every even number is a sum of at most six primes (Ramaré, 1995) and every
odd number is a sum of at most five (Tao, 2012).
< / p >
< center > < img src = "Letter.jpg" > < / center >
< p class = "inwebparagraph" > Computer verification has been made up to around \(10^{18}\), but by rather better
methods.
< / p >
< pre class = "definitions" >
< span class = "definitionkeyword" > define< / span > < span class = "constant" > RANGE< / span > < span class = "plain" > < / span > < span class = "constant" > 100< / span >
< / pre >
< pre class = "display" >
< 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 >
< 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 >
< 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 >
< < span class = "cwebmacro" > Solve Goldbach's conjecture for i< / span > < span class = "cwebmacronumber" > 1.1< / span > > < span class = "character" > ;< / span >
< span class = "plain" > }< / span >
< / pre >
< p class = "inwebparagraph" > < / p >
< p class = "inwebparagraph" > < a id = "SP1_1" > < / a > < b > § 1.1. < / b > This ought to print:
< / p >
< pre class = "display" >
< span class = "plain" > 4 = 2+2< / span >
< span class = "plain" > 6 = 3+3< / span >
< 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 >
< / pre >
< p class = "inwebparagraph" > We'll print each different pair of primes adding up to i. We
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 >
< p class = "macrodefinition" > < code class = "display" >
< < span class = "cwebmacrodefn" > Solve Goldbach's conjecture for i< / span > < span class = "cwebmacronumber" > 1.1< / span > > =
< / code > < / p >
< 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 >
< 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 >
< / pre >
< p class = "inwebparagraph" > < / p >
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< p class = "endnote" > This code is used in < a href = "S-sp.html#SP1" > § 1< / a > .< / p >
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< hr class = "tocbar" >
< 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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