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Lim sup of difference of primes

  1. Aug 10, 2006 #1
    Although Andrica's conjecture is still unsolved, I'm told that it is possible to prove that

    [tex]\lim\sup_{n\rightarrow\infty}\sqrt{p_{n+1}}-\sqrt{p_n}=1[/tex].

    Does anyone know how or can point me to a source?
     
  2. jcsd
  3. Aug 10, 2006 #2

    shmoe

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    Who told you that was true? It looks an awful lot like the max occurs when n=4, so the primes 7 and 11 and seems to decrease from there. I've seen it conjectured that the full limit is actually zero, not much of a conjecture if the lim sup was known to be 1.
     
  4. Aug 10, 2006 #3

    CRGreathouse

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    I can't imagine the limit being higher than 0. Heck, find a number that makes it go higher than 0.01 for n > 10^9 and I'll be suprised.
     
  5. Aug 12, 2006 #4
    Yes, that was a typo. I meant 0.
     
  6. Aug 12, 2006 #5

    shmoe

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    If the lim sup was 0, then the limit would be 0. This was still an unsolved problem according to Guy's 2004 "unsolved problems in number theory".

    Maybe they meant

    [tex]\lim\inf_{n\rightarrow\infty}\sqrt{p_{n+1}}-\sqrt{p_n}=0[/tex]

    which you can manage. Use the fact that [tex]p_{n+1}-p_{n}\leq \log p_n[/tex] is true infinitely often (much more is true actually, see Goldstom, Pintz and Yildrims recent work).
     
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