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< / ul > < h2 > Foundation Module< / h2 > < ul >
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<!-- Weave of 'Summing Primes' generated by Inweb -->
< 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 > < 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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< / 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 >
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< p class = "inwebparagraph" > Besides which, < span class = "extract" > < span class = "extract-syntax" > printf(k+1)< / span > < / span > , to say the least: see < a href = "http://www.google.com" class = "external" > http://www.google.com< / a >
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or for that matter see < a href = "S-tsoe.html" class = "internal" > The Sieve of Eratosthenes< / a > .
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< / p >
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< center > < img src = "Letter.jpg" height = "720" > < / center >
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< p class = "inwebparagraph" > Computer verification has been made up to around \(10^{18}\), but by rather better
methods.< sup id = "fnref:1" > < a href = "#fn:1" rel = "footnote" > 1< / a > < / sup > Which is awesome.< sup id = "fnref:2" > < a href = "#fn:2" rel = "footnote" > 2< / a > < / sup > And \(i< j\), that's for sure.
$$ \int_0^1 \cos x {\rm d}x $$
Which is really the point.
< / p >
< 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 > And don't just take my word for it.
< 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 > Really!
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< a href = "#fnref:2" title = "return to text" > ↩ < / a > < / p > < / li > < / ul > < p class = "inwebparagraph" > < a id = "SP2" > < / a > < b > § 2. < / b > And now...
< / p >
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< div class = "carousel-container" id = "carousel-no-1" >
< div class = "carousel-slide fading-slide" style = "display: block;" >
< div class = "carousel-caption-above" > This is numero 1< / div >
< div class = "carousel-number-above" > 1 / 3< / div >
< div class = "carousel-content" > < center > < img src = "img1.jpg" > < / center >
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< / div >
< / div >
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< div class = "carousel-slide fading-slide" style = "display: none;" >
< div class = "carousel-number" > 2 / 3< / div >
< div class = "carousel-content" > < center > < img src = "img2.jpg" > < / center >
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< / div >
< / div >
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< div class = "carousel-slide fading-slide" style = "display: none;" >
< div class = "carousel-number-below" > 3 / 3< / div >
< div class = "carousel-content" > < center > < img src = "img3.jpg" > < / center >
< / div >
< div class = "carousel-caption-below" > So, this is a program to see if even numbers from 4 to 100 can all be written as a sum of two primes. Christian Goldbach asked Euler in 1742 if every even number greater than 2 can be written this way.< / div >
< / div >
< a class = "carousel-prev-button" onclick = "carouselMoveSlide("carousel-no-1", "carousel-dots-no-1", -1)" > ❮ < / a >
< a class = "carousel-next-button" onclick = "carouselMoveSlide("carousel-no-1", "carousel-dots-no-1", 1)" > ❯ < / a >
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< span class = "carousel-dot" onclick = "carouselSetSlide("carousel-no-1", "carousel-dots-no-1", 2)" > < / span >
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< / div >
< p class = "inwebparagraph" > So there, really!
< / p >
< p class = "inwebparagraph" > < a id = "SP3" > < / a > < b > § 3. < / b > < / p >
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< pre class = "definitions" >
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< 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" >
< 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 >
< 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" > stepping in twos to stay even< / span >
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< span class = "plain-syntax" > < / span > < < span class = "named-paragraph" > Solve Goldbach's conjecture for i< / span > < span class = "named-paragraph-number" > 3.1< / span > > < span class = "character-syntax" > ;< / span >
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< span class = "character-syntax" > }< / span >
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< / pre > < p class = "inwebparagraph" > < a id = "SP3_1" > < / a > < b > § 3.1. < / b > This ought to print:
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< / p >
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< pre class = "ConsoleText-displayed-code all-displayed-code" >
< 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 > < span class = "ConsoleText-identifier-syntax" > -fish< / span > < span class = "ConsoleText-plain-syntax" > 1< / span >
< span class = "ConsoleText-plain-syntax" > 4 = 2+2< / span >
< span class = "ConsoleText-plain-syntax" > 6 = 3+3 < / span > < a href = "https://www.wikipedia.org" class = "external" > https://www.wikipedia.org< / a >
< 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 >
< 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" >
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< < span class = "named-paragraph-defn" > Solve Goldbach's conjecture for i< / span > < span class = "named-paragraph-number" > 3.1< / span > > =
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< / code > < / p >
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< pre class = "displayed-code all-displayed-code" >
< 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 > < ul class = "endnotetexts" > < li > This code is used in < a href = "S-sp.html#SP3" > § 3< / a > .< / li > < / ul > < hr class = "tocbar" >
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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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