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primes between squares calcs

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  • Bill
    Aah! It s good to see my favorite math problem, the number of primes between squares, reappear on primenumbers, and while it is up, I would like to mention
    Message 1 of 5 , Dec 8, 2010
      Aah! It's good to see my favorite math problem, the number of primes between squares, reappear on primenumbers, and while it is up, I would like to mention Capelle's wondeful conjecture A, posted here on primenumbers Sept 4, 2005, message number 16999. It is an extension of Legendre's conjecture, which he uses as a basis for his expansive formula.
      Capelle conjecture A is:
      Pi((m+1)^n) - Pi(m^n) >= m^(n-2) for n>=2, m>=1 .
      Legendre's conjecture claims there is at least one prime between the squares of consecutive integers, and is Capelle's case with n=2 and m = 1.

      I have some ideas how Capelle's conjecture can be made even more powerful, and I have done several handfulls of calculations to check the viability of my approach, but I of course need to do a lot more calculations. Can anyone direct me to a database that tells me the number of primes between two positive integers, or perhaps has someone checked out Capelle's conjecture A to any significant degree? Any leads or previous work would lighten my task.

      Thanks, Bill Oscarson
    • Patrick
      ... http://tech.groups.yahoo.com/group/primenumbers/message/17020 Patrick Capelle
      Message 2 of 5 , Dec 9, 2010
        --- In primenumbers@yahoogroups.com, "Bill" <bill2math@...> wrote:
        >
        > Aah! It's good to see my favorite math problem, the number of primes between squares, reappear on primenumbers, and while it is up, I would like to mention Capelle's wondeful conjecture A, posted here on primenumbers Sept 4, 2005, message number 16999. It is an extension of Legendre's conjecture, which he uses as a basis for his expansive formula.
        > Capelle conjecture A is:
        > Pi((m+1)^n) - Pi(m^n) >= m^(n-2) for n>=2, m>=1 .
        > Legendre's conjecture claims there is at least one prime between the squares of consecutive integers, and is Capelle's case with n=2 and m = 1.
        >
        > I have some ideas how Capelle's conjecture can be made even more powerful, and I have done several handfulls of calculations to check the viability of my approach, but I of course need to do a lot more calculations. Can anyone direct me to a database that tells me the number of primes between two positive integers, or perhaps has someone checked out Capelle's conjecture A to any significant degree? Any leads or previous work would lighten my task.
        >
        > Thanks, Bill Oscarson
        >


        http://tech.groups.yahoo.com/group/primenumbers/message/17020

        Patrick Capelle
      • Bill
        OOPs! I left out a in my definition of Legendre s conjecture. Legendre s conjecture is of course for n=2 and m = 1. My apologies Bill
        Message 3 of 5 , Dec 9, 2010
          OOPs! I left out a ">" in my definition of Legendre's conjecture.
          Legendre's conjecture is of course for n=2 and m >= 1.
          My apologies Bill
        • Aldrich
          ... Intuitively Legendre s conjecture would appear to be true: the rate of growth of the gap between squares looks like it would far exceed the rate of growth
          Message 4 of 5 , Dec 12, 2010
            --- In primenumbers@yahoogroups.com, "Bill" <bill2math@...> wrote:
            >

            > Legendre's conjecture claims there is at least one prime between the squares of consecutive integers, and is Capelle's case with n=2 and m = 1.
            >

            Intuitively Legendre's conjecture would appear to be true: the
            rate of growth of the gap between squares looks like it would
            far exceed the rate of growth in the average gap between primes.
            However the pattern of these gaps can be very uneven and I believe that
            it HAS been proven that the gap between primes can be arbitrarily
            large. This does not disprove Legendre's conjecture - probably
            the only way to do that would be to find a counterexample. Good
            with that project!

            a.
          • Paul Leyland
            ... The proof is trivial. For prime p, all integers n s.t. p!+1
            Message 5 of 5 , Dec 12, 2010
              On Sun, 2010-12-12 at 10:13 +0000, Aldrich wrote:
              > However the pattern of these gaps can be very uneven and I believe
              > that
              > it HAS been proven that the gap between primes can be arbitrarily
              > large. This does not disprove Legendre's conjecture - probably

              The proof is trivial. For prime p, all integers n s.t.
              p!+1 < n <= p!+p are divisible by at least one prime <= p and p can be
              taken arbitrarily large.

              Paul
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