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"problem" with Euclid's proof of the Infinitude of Primes

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  • duke nukem
    I am having a problem with Euclid s proof of the Infinitude of Primes. Sorry for the not so short post ahead :) Please see this link for the modern
    Message 1 of 1 , Jan 31, 2004
      I am having a "problem" with Euclid's proof of the Infinitude of Primes.

      Sorry for the 'not so short post' ahead :)

      Please see this link for the modern translation of the proof (note there are 2 versions there, don't forget the one in the bottom of the page :).
      These versions are considered true (of course) and can be found in most textbooks.
      the link:
      http://primes.utm.edu/notes/proofs/infinite/euclids.html

      Now the "problem" is that I came across this "ShortCut Version" of proof which looks like a real shortcut to Euclid's proof.
      I am wondering if there is any mistake/hidden assumption/etc. I am missing here:
      The proof gos like that:
      "
      (1). The prime numbers are the numbers p1,p2,...,pn,... of set S.
      Suppose S is finite [supposition].

      (2). If S is finite then: p1,p2,...,pn is the complete series set, so that pn is the largest prime.

      (3). Form W+1 = (p1xp2x,...,xpn) + 1.
      W+1 is not in the set S so it is not prime.
      W+1 is not divisible by any of primes in S_P so it
      is prime.
      Contradiction!.
      Reverse supposition and primes are infinite.
      "

      P.S.
      I do not think the 'counter example': 2*3*5*7*11*13+1= 30031 = 59*509 => hence a composite W+1, work here.
      Why?
      because 2,3,5,7,11,13 isn't the Actual list of All primes. its a partial list and for that reason doesn't necessarily has the same attribute as the complete list.
      To me this looks the same as this:
      say 1,2,3,4,5 are all the "favorite" numbers.
      ["favorite" - a set I just made up].
      My Great Theorem: the sum of all "favorite" numbers is odd.
      proof: 1+2+3+4+5=15. QED :)
      Now one would say: "but look - 1+2+3=6" so there is a partial sum that is Even.
      And I would answer: but my theorem is about the sum of ALL the favorite numbers, so we are talking about different cases!.
      * would I be correct? If not, why?


      Thanks for your kind insights,
      -Ben
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