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13949Re: [PrimeNumbers] Infinite primes-> a Turing Machine prime sieve that never stops?

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  • Andy Swallow
    Nov 4, 2003
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      > But there have to be two schools of thought
      > since "proof" isn't working.
      > 1) Euclid's school infinite primes exist
      > 2) modern school of thought : infinite primes don't exist
      >
      > Unless you can "conclusively" prove no infinite prime exists...
      > I've seen nothing like that in the posts.

      Make up your mind Roger, what are you arguing? Are you (a) talking about
      whether the number of primes is infinite or not, or (b) whether there are
      some magical new numbers which you decide to call "infinite primes"?
      You're quite correct of course, nothing in the posts has proved that no
      infinite prime exists. In the case of (a), your attempts at proof have
      been less than wonderful, and in the case of (b) none of us have any
      idea what you mean by infinite prime.

      I presume you're talking about whether there are infinitely many primes
      or not, but you still seem to think that this would imply the existence
      of "infinitely large primes". Not true. There would be *arbitrarily*
      large primes, but they would always be finite. It's important to
      understand this. Don't you have any number theory books you can look at
      for this stuff? Try the first chapter of an introductory book...

      As a little amusement, suppose that there were only finitely many
      primes. Then the Euler product form of the zeta function would define an
      entire function, thus making zeta(s) an entirely different animal.
      Congratulations Roger, you would have answered the Riemann hypothesis, one
      way or another.

      Andy

      PS "Modern school of thought"? You mean "Roger's school of thought"?
      Small school then?
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