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

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  • Jose Ramón Brox
    ... [...] ... pn is the largest prime. ... You are right when you say W+1 is not in the set, but are wrong when you say it has to be prime. You are missing one
    Message 1 of 1 , Feb 1, 2004
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      ----- Original Message -----
      [...]
      > (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.
      > "

      You are right when you say W+1 is not in the set, but are wrong when you say
      it has to be prime. You are missing one case:

      If W+1 is not divisible by any prime in S then automatically you have to
      deduce that S is not the set of all primes: then you have to admit that any
      number bigger than pn can be prime as well; so W+1 is either prime or
      divisible by primes between pn and W+1.

      Jose Brox.
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