document weave order 0
  head banner <Weave of 'Complete Program' generated by Inweb>
  body
    chapter <Sections>
      chapter header <Sections>
      section <Summing Primes>
        section header <Summing Primes>
        section purpose <Here we verify the conjecture for small numbers.>
        paragraph P1
          material discussion
            commentary <On 7 June 1742, Christian Goldbach wrote a letter from Moscow to Leonhard\n>
            commentary <Euler in Berlin making "eine conjecture hazardiren" that every even number\n>
            commentary <greater than >
            mathematics <2>
            commentary < can be written as a sum of two primes.>
            footnote_cue [1]
            commentary < Euler did not\n>
            commentary <know if this was true, and nor does anyone else.\n>
            figure <Letter.jpg> -1 by 720
            commentary <Goldbach, a professor at St Petersburg and tutor to Tsar Peter II, wrote in\n>
            commentary <several languages in an elegant cursive script, and was much valued as a\n>
            commentary <letter-writer, though his reputation stands less high today.>
            footnote_cue [2]
            commentary < All the same,\n>
            commentary <the general belief now is that primes are just plentiful enough, and just\n>
            commentary <evenly-enough spread, for Goldbach to be right. It is known that:\n>
            item depth 1 label <a>
            commentary <every even number is a sum of at most six primes (Ramaré, 1995), and\n>
            item depth 1 label <b>
            commentary <every odd number is a sum of at most five (Tao, 2012).\n>
          material footnotes
            footnote [1]
              footnote_cue [1]
              commentary < "Greater than 2" is our later proviso: Goldbach needed no such exception\n>
              commentary <because he considered 1 a prime number, as was normal then, and was sometimes\n>
              commentary <said as late as the early twentieth century.\n>
            footnote [2]
              footnote_cue [2]
              commentary < Goldbach, almost exactly a contemporary of Voltaire, was a good citizen\n>
              commentary <of the great age of Enlightenment letter-writing. He and Euler exchanged\n>
              commentary <scholarly letters for over thirty years, not something Euler would have\n>
              commentary <kept up with a duffer. Goldbach was also not, as is sometimes said, a lawyer.\n>
              commentary <See: >
              url content <http://mathshistory.st-andrews.ac.uk/Biographies/Goldbach.html> url <http://mathshistory.st-andrews.ac.uk/Biographies/Goldbach.html>
              commentary <.\n>
              commentary <An edited transcription of the letter is at: >
              url content <http://eulerarchive.maa.org//correspondence/letters/OO0765.pdf> url <http://eulerarchive.maa.org//correspondence/letters/OO0765.pdf>
              commentary <\n>
        paragraph P2
          material discussion
            commentary <Computer verification has been made up to around >
            mathematics <10^{18}>
            commentary <, but by rather\n>
            commentary <better methods than the one we use here. We will only go up to:\n>
          material definition
            code line
              defn <define>
              source_code <RANGE 100>
                          _nnnnnpnnn_
          material code: C
            code line
              source_code <#include <stdio.h>>
                          _piiiiiiippiiiiipip_
            vskip
            code line
              source_code <int main(int argc, char *argv[]) {>
                          _rrrpffffprrrpiiiipprrrrppiiiippppp_
            code line
              source_code <    for (int i=4; i<RANGE; i=i+2) >
                          _pppprrrpprrrpippppipnnnnnppipipppp_
              commentary < stepping in twos to stay even> (code)
            code line
              source_code <        >
                          _pppppppp_
              pmac <Solve Goldbach's conjecture for i>
              source_code <;>
                          _c_
            code line
              source_code <}>
                          _c_
        paragraph P2.1
          material discussion
            commentary <This ought to print:\n>
          material code: ConsoleText
            code line
              source_code <    $ goldbach/Tangled/goldbach>
                          _ppppepfffffffffffffffffffffffff_
            code line
              source_code <    4 = 2+2>
                          _ppppppppppp_
            code line
              source_code <    6 = 3+3>
                          _ppppppppppp_
            code line
              source_code <    8 = 3+5>
                          _ppppppppppp_
            code line
              source_code <    10 = 3+7 = 5+5>
                          _pppppppppppppppppp_
            code line
              source_code <    12 = 5+7>
                          _pppppppppppp_
            code line
              source_code <    14 = 3+11 = 7+7>
                          _ppppppppppppppppppp_
            code line
              source_code <    ...>
                          _ppppppp_
          material discussion
            commentary <We'll print each different pair of primes adding up to >
            mathematics <i>
            commentary <. We\n>
            commentary <only check in the range >
            mathematics <2 \leq j \leq i/2>
            commentary < to avoid counting pairs\n>
            commentary <twice over (thus >
            mathematics <8 = 3+5 = 5+3>
            commentary <, but that's hardly two different ways).\n>
          material paragraph macro
            code line
              pmac <Solve Goldbach's conjecture for i> (definition)
          material code: C
            code line
              source_code <    printf("%d", i);>
                          _ppppiiiiiipssssppipp_
            code line
              source_code <    for (int j=2; j<=i/2; j++)>
                          _pppprrrpprrrpippppippippppippp_
            code line
              source_code <        if ((>
                          _pppppppprrppp_
              function usage <isprime>
              source_code <(j)) && (>
                          _pippppppp_
              function usage <isprime>
              source_code <(i-j)))>
                          _pipippp_
            code line
              source_code <            printf(" = %d+%d", j, i-j);>
                          _ppppppppppppiiiiiipssssssssssppippipipp_
            code line
              source_code <    printf("\n");>
                          _ppppiiiiiipsssspp_
          material endnotes
            endnote
              commentary <This code is >
              commentary <used in >
              locale P2
              commentary <.>
        section footer <Summing Primes>
      section <The Sieve of Eratosthenes>
        section header <The Sieve of Eratosthenes>
        section purpose <A fairly fast way to determine if small numbers are prime, given storage.>
        toc - <S/tsoe>
          toc line - <S1, Storage> P1'Storage'
          toc line - <S2, Primality> P2'Primality'
        paragraph P1'Storage'
          material discussion
            commentary <This technique, still essentially the best sieve for finding prime\n>
            commentary <numbers, is attributed to Eratosthenes of Cyrene and dates from the 200s BC.\n>
            commentary <Since composite numbers are exactly those numbers which are multiples of\n>
            commentary <something, the idea is to remove everything which is a multiple: whatever\n>
            commentary <is left, must be prime.\n>
            vskip (in comment)
            commentary <This is very fast (and can be done more quickly than the implementation\n>
            commentary <below), but (a) uses storage to hold the sieve, and (b) has to start right\n>
            commentary <back at 2 - so it can't efficiently test just, say, the eight-digit numbers\n>
            commentary <for primality.\n>
          material code: C
            code line
              source_code <int still_in_sieve[RANGE + 1];>
                          _rrrpiiiiiiiiiiiiiipnnnnnpppnpp_
            code line
              source_code <int sieve_performed = FALSE;>
                          _rrrpiiiiiiiiiiiiiiipppnnnnnp_
        paragraph P2'Primality'
          material discussion
            commentary <We provide this as a function which determines whether a number is prime:\n>
          material definition
            code line
              defn <define>
              source_code <TRUE 1>
                          _nnnnpn_
            code line
              defn <define>
              source_code <FALSE 0>
                          _nnnnnpn_
          material code: C
            code line
              source_code <int >
                          _rrrp_
              function defn <isprime>
                commentary <Summing Primes>
                commentary < - >
                locale P2.1
              source_code <(int n) {>
                          _prrrpippp_
            code line
              source_code <    if (n <= 1) return FALSE;>
                          _pppprrppippppnpprrrrrrpnnnnnp_
            code line
              source_code <    if (n > RANGE) { printf("Out of range!\n"); return FALSE; }>
                          _pppprrppipppnnnnnppppiiiiiipsssssssssssssssssppprrrrrrpnnnnnppp_
            code line
              source_code <    if (!sieve_performed) >
                          _pppprrpppiiiiiiiiiiiiiiipp_
              pmac <Perform the sieve>
              source_code <;>
                          _p_
            code line
              source_code <    return still_in_sieve[n];>
                          _pppprrrrrrpiiiiiiiiiiiiiipipp_
            code line
              source_code <}>
                          _p_
        paragraph P2.1
          material discussion
            commentary <We save a little time by noting that if a number up to >
            inline
              source_code <RANGE>
                          _xxxxx_
            commentary < is composite\n>
            commentary <then one of its factors must be smaller than the square root of >
            inline
              source_code <RANGE>
                          _xxxxx_
            commentary <. Thus,\n>
            commentary <in a sieve of size 10000, one only needs to remove multiples of 2 up to 100,\n>
            commentary <for example.\n>
          material paragraph macro
            code line
              pmac <Perform the sieve> (definition)
          material code: C
            code line
              source_code <    >
                          _pppp_
              pmac <Start with all numbers from 2 upwards in the sieve>
              source_code <;>
                          _p_
            code line
              source_code <    for (int n=2; n*n <= RANGE; n++)>
                          _pppprrrpprrrpippppipippppnnnnnppippp_
            code line
              source_code <        if (still_in_sieve[n])>
                          _pppppppprrppiiiiiiiiiiiiiipipp_
            code line
              source_code <            >
                          _pppppppppppp_
              pmac <Shake out multiples of n>
              source_code <;>
                          _p_
            code line
              source_code <    sieve_performed = TRUE;>
                          _ppppiiiiiiiiiiiiiiipppnnnnp_
          material endnotes
            endnote
              commentary <This code is >
              commentary <used in >
              locale P2'Primality'
              commentary <.>
        paragraph P2.1.1
          material paragraph macro
            code line
              pmac <Start with all numbers from 2 upwards in the sieve> (definition)
          material code: C
            code line
              source_code <    still_in_sieve[1] = FALSE;>
                          _ppppiiiiiiiiiiiiiippppppnnnnnp_
            code line
              source_code <    for (int n=2; n <= RANGE; n++) still_in_sieve[n] = TRUE;>
                          _pppprrrpprrrpippppippppnnnnnppippppiiiiiiiiiiiiiipippppnnnnp_
          material endnotes
            endnote
              commentary <This code is >
              commentary <used in >
              locale P2.1
              commentary <.>
        paragraph P2.1.2
          material paragraph macro
            code line
              pmac <Shake out multiples of n> (definition)
          material code: C
            code line
              source_code <    for (int m= n+n; m <= RANGE; m += n) still_in_sieve[m] = FALSE;>
                          _pppprrrpprrrpippipippippppnnnnnppippppippiiiiiiiiiiiiiipippppnnnnnp_
          material endnotes
            endnote
              commentary <This code is >
              commentary <used in >
              locale P2.1
              commentary <.>
        section footer <The Sieve of Eratosthenes>
      chapter footer <Sections>
  tail rennab <End of weave>