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Possible Proof?

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  • David Cleaver
    Hello all, I m not sure how to write a formal proof, but I think I may have something worth looking at. Now, in order to prove that there is at least one
    Message 1 of 3 , Jan 3, 2004
      Hello all,

      I'm not sure how to write a formal proof, but I think
      I may have something worth looking at. Now, in order
      to prove that there is at least one prime between x^2
      and (x+1)^2, I believe it suffices to show that we need
      at least one prime between x and x + x^(1/2).

      Now, my idea is to use a lower bound on the prime counting
      function, pi(x), and then use that to prove existence.
      So, for my lower bound I choose pi(x) > x^(2/3).
      Now, I know that x^(2/3) is a small lower bound, but I do
      _believe_ that pi(x) is _always_ greater than it. So, since
      there are always at least x^(2/3) primes less than x, we just
      need the difference of pi(x+x^(1/2)) and pi(x) to be greater
      than one, which tells us there is at least one prime in that
      interval.

      So, replacing pi(x+x^(1/2)) with (x+x^(1/2))^(2/3)
      and replacing pi(x) with x^(2/3)
      We see that the difference of these two is:
      (x+x^(1/2))^(2/3) - x^(2/3) > 1 for all x >= 15

      Now, I realize there may be flaws in my logic, or that I used
      too loose a bound, but please be gentle as I'm only suggesting
      a possible proof and not an absolute EUREKA! After that I'd
      like to know what everyone thinks. Thanks for your time.

      -David C.
    • Andrew Swallow
      Your idea is to study pi(x+x^(1/2))-pi(x), using a lower bound for pi(x). But to get a lower bound for pi(x+x^(1/2))-pi(x), you need a lower bound for
      Message 2 of 3 , Jan 3, 2004
        Your idea is to study pi(x+x^(1/2))-pi(x), using a lower bound for
        pi(x). But to get a lower bound for pi(x+x^(1/2))-pi(x), you need a
        lower bound for pi(x+x^(1/2)) and an *upper* bound for pi(x). You
        can't use the lower bound for both.

        Andy

        --- In primenumbers@yahoogroups.com, David Cleaver <wraithx@m...>
        wrote:
        > Hello all,
        >
        > I'm not sure how to write a formal proof, but I think
        > I may have something worth looking at. Now, in order
        > to prove that there is at least one prime between x^2
        > and (x+1)^2, I believe it suffices to show that we need
        > at least one prime between x and x + x^(1/2).
        >
        > Now, my idea is to use a lower bound on the prime counting
        > function, pi(x), and then use that to prove existence.
        > So, for my lower bound I choose pi(x) > x^(2/3).
        > Now, I know that x^(2/3) is a small lower bound, but I do
        > _believe_ that pi(x) is _always_ greater than it. So, since
        > there are always at least x^(2/3) primes less than x, we just
        > need the difference of pi(x+x^(1/2)) and pi(x) to be greater
        > than one, which tells us there is at least one prime in that
        > interval.
      • richard042@yahoo.com
        I wrote a paper covering some aspects of this problem a while ago. To this day, no one who looked at it or received it has said anything to me about it, but I
        Message 3 of 3 , Jan 3, 2004
          I wrote a paper covering some aspects of this problem a while ago.
          To this day, no one who looked at it or received it has said anything
          to me about it, but I certainly would appreciate that. The Less Than
          Double Theorem states, in essence, that if you sum the number of
          divisors of each of the first 2x integers, then double the result and
          subtract x, you get about the sum of the number of divisors of the 2x
          integers between x^2 and (x+1)^2.
          The link is

          http://www.imathination.net/D_Boland_V2.PDF

          It establishes upper and lower bounds on the sum of the number of
          divisors of the (2n+1) consecutive integers (n^2+1),...,(n+1)^2. The
          theorem relates that summation to the same summation based over the
          first 2n+1 integers {1,2,...,2n+1}. The curve of the actual values
          approaches a constant between the two curves of the upper and lower
          bounds. c=0.92... such that (actuals ~= lower bound + 2nc)

          The constant is probably a natural result related to the Euler-
          Mascheroni constant, it certainly needs to be fleshed out and
          investigated further, but I hit a roadblock on converging the
          constant. The technique in the theorem can be used to create other
          divisor sum models with different spans or perhaps modular
          restrictions as might be needed for application to other problems or
          continued work on this problem. I haven't looked at this in awhile,
          but I do recall thinking the bounds could likely be tightened quite a
          bit more with some quadratic residue theory applied, we'll see.

          Relating divisors to primes is the next big step, anyone got any
          ideas or appropriate references?

          -Dick
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