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Re: [PrimeNumbers] LI(Pn#)=P(n-1)# (approximately)

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  • Phil Carmody
    ... What s the size of Pn#? (See http://primes.utm.edu/glossary/page.php?sort=Primorial ) Therefore what s the expected density of primes around an arbitrary
    Message 1 of 2 , Mar 15, 2007
      --- Wes & Paige Bullock <6bullocks@...> wrote:
      > I found something interesting when I was investigating the density of primes
      > near the large prime gaps around the primorials.
      >
      > Basically, it could be stated as the following conjecture:
      >
      > The number of primes less than a primorial value, Pn#, approaches P(n-1)#
      > (in a percentage error sense) as n increases.

      What's the size of Pn#? (See
      http://primes.utm.edu/glossary/page.php?sort=Primorial )

      Therefore what's the expected density of primes around an arbitrary number in
      the vicinity of Pn#?

      There's nothing special about primorials in this regard, they behave exactly as
      arbitrary numbers of similar size.

      Phil

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    • Werner D. Sand
      Proof: You don t need Li, PNT is enough. We start with a form of PNT: sum(ln p)(p lim(n- inf)(sum(ln p)(p lim(n- inf)(pn /
      Message 2 of 2 , Mar 16, 2007
        Proof:

        You don't need Li, PNT is enough. We start with a form of PNT:
        sum(ln p)(p<=pn) ~ pn. ==>
        lim(n->inf)(sum(ln p)(p<=pn) / pn) = 1 ==>
        lim(n->inf)(pn / sum(ln(p)(p<=pn) = 1 ==>
        lim(n->inf)(pn / ln(pn#)(p<=pn) = 1 ==>
        lim(n->inf)(pn*p(n-1)#) / (ln(pn#)*p(n-1)#) = 1 ==>
        lim(n->inf)(pn#) / (ln(pn#)*p(n-1)#) ) = 1 ==>
        lim(n->inf)(pn#) / ln(pn#) = p(n-1)# = PNT, qed.

        Werner
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