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21442Re: product convergence

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  • djbroadhurst
    May 8, 2010
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      --- In primenumbers@yahoogroups.com,
      Kermit Rose <kermit@...> wrote:

      > what is wrong with my "Proof"

      It began OK and then fizzled out.
      Here is an elementary proof.

      Proposition: The product prod(p>2, (p-2)/p) over primes p>2 vanishes.

      Proof: It suffices to show that the contrary proposition is absurd.
      Suppose that prod(p>2, (p-2)/p) did not vanish.
      Then, by taking logs, we would conclude that
      sum(p>2, log(p) - log(p-2)) is finite. But
      log(p) - log(p-2) > 2/p. Thus
      sum(p>2, 1/p) would also be finite.
      Yet that is easily proven to be absurd:
      http://primes.utm.edu/infinity.shtml#punchline
      Hence the product prod(p>2, (p-2)/p) does indeed vanish.

      Comment: By the same argument, it follows that
      the product prod(p>x, (p-x)/p) over primes p > x
      vanishes for all real x > 0.

      David
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