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Re: reciprocal consecutive primes

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  • jbrennen
    ... I think this is implied by the Iwaniec & Pintz finding, namely that there is always a prime between x and x-x^(23/42), for any real number x 11. The
    Message 1 of 5 , Nov 3, 2005
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      --- theo2357 wrote:
      >
      > Let p,q,r be three consecutive primes. How can be proven that
      > 1/p < 1/q + 1/r? It's NOT trivial!
      >

      I think this is implied by the Iwaniec & Pintz finding, namely
      that there is always a prime between x and x-x^(23/42), for
      any real number x > 11.

      The Iwaniec & Pintz theorem can be used to show the weaker result,
      that for prime p >= 5, nextprime(p)/p < Phi, where Phi is the
      golden ratio, (sqrt(5)+1)/2 = 1.618...

      And if q/p < Phi, and r/q < Phi, then the original inequality
      1/p < 1/q + 1/r follows quite easily.
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