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6213Twin primes was: Infinite primes

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  • David Litchfield
    Mar 30, 2002
      > This is purported to be the original, although I fail to see why -1 wasn't
      > used instead.

      Yep. The reason I ask was because this could prove the twin prime
      conjecture.

      If q= P1 * P2 * ... Pn + 1

      then q is prime because q mod and Pn is 1.
      Using the same logic, it also follows that

      r = P1 * P2 * .... Pn -1

      is also prime because r mod any Pn equals Pn-1

      q - r = 2.

      What's more as twin primes must be +1 or -1 an even multiple of 3 (due to
      the location and frequency of multiples of three : the only possible place
      for twin primes is either side of 2*3*n) then P1 * P2 * .... Pn should be an
      even multiple of 6 if indeed r and q are twin primes.

      As P1 = 2 and P2 = 3 then P1*P2*...Pn must be an even multiple of 3 - so
      fulfulling the "twin primeness" of r and q.

      That said in _reality_ q and r could both be composite - with two or more
      primes not in P1 to Pn being the factors. But then this is confusing reality
      with a hypothetical situtation so does this proof for the twin prime
      conjecture stand in the same way the Euclid's proof is accepted. At best
      this proves the twin prime conjecture - at worst it proves at least the
      possibility of an infinite number of twin primes.

      TIA,
      David Litchfield




      ----- Original Message -----
      From: "Jon Perry" <perry@...>
      To: "Prime Numbers" <primenumbers@yahoogroups.com>
      Sent: Saturday, March 30, 2002 9:12 AM
      Subject: RE: [PrimeNumbers] Infinite primes


      >
      > There is an interesting paper that covers various proofs of the infinitude
      > of primes at:
      >
      > http://algo.inria.fr/banderier/Seminar/Vardi/index.html
      >
      > Jon Perry
      > perry@...
      > http://www.users.globalnet.co.uk/~perry/maths
      > BrainBench MVP for HTML and JavaScript
      > http://www.brainbench.com
      >
      >
      > -----Original Message-----
      > From: David Litchfield [mailto:Mnemonix@...]
      > Sent: 29 March 2002 22:40
      > To: Prime Numbers
      > Subject: [PrimeNumbers] Infinite primes
      >
      >
      > I know Euclid proved there were an infinite number of primes. Was this his
      > proof?
      >
      > Assume there are a finite number of primes - P1, P2, P3....Pn
      >
      > q = P1 * P2 * P3 * .... Pn + 1
      >
      > q divided by any of Px always leaves a remainder of 1 and therefore q must
      > also be prime and as q was not one of P1 to Pn then there are an infinite
      > number of primes.
      >
      > I know Ribenboim and Kummer produced proofs that were variants of Euclid's
      > proof - I'm just wondering if the above was the original?
      >
      > TIA,
      > David
      >
      >
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