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Primes between squares

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  • Jon Perry
    Is there always a prime between x^2 and (x+n)^2? Jon Perry perry@globalnet.co.uk http://www.users.globalnet.co.uk/~perry Brainbench Most Valuable
    Message 1 of 8 , May 18, 2001
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      Is there always a prime between x^2 and (x+n)^2?

      Jon Perry
      perry@...
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    • Ron McCluskey
      Is there always a prime between x^2 and (x+n)^2? Jon Perry A casual glance would suggest that there is. Furthermore, it would seem that the number of primes
      Message 2 of 8 , May 18, 2001
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        Is there always a prime between x^2 and (x+n)^2?

        Jon Perry

        A casual glance would suggest that there is. Furthermore, it would seem that the number of primes between x^2 and (x+1)^2 increases as x increases. I ran through the cases from x=1 to x=19. The number of primes between starts out with an average of 2 and ends with an average of 6 when x=19.
        Ron McCluskey



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      • jack@brennen.net
        ... The answer is almost certainly yes, but it s unproven. See the last entry on the page: http://www.utm.edu/research/primes/notes/conjectures/
        Message 3 of 8 , May 18, 2001
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          --- In primenumbers@y..., "Jon Perry" <perry@g...> wrote:
          > Is there always a prime between x^2 and (x+n)^2?
          >

          The answer is almost certainly yes, but it's unproven.

          See the last entry on the page:

          http://www.utm.edu/research/primes/notes/conjectures/
        • Jon Perry
          Thanks. I just wondered if the easier case had been solved for any n, including functions, etc... For example, is there always a prime between x^2 and (x+x)^2,
          Message 4 of 8 , May 20, 2001
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            Thanks. I just wondered if the easier case had been solved for any n,
            including functions, etc...

            For example, is there always a prime between x^2 and (x+x)^2, or even
            (x^x)^2.

            Someone stated that this was not true for n=0, which is true. As it is true
            for n=1 (supposedly), this would imply that there exists a real y such that
            y is minimum and there is always a prime between x^2 and (x+y)^2.

            Jon Perry
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            -----Original Message-----
            From: jack@... [mailto:jack@...]
            Sent: 18 May 2001 20:43
            To: primenumbers@yahoogroups.com
            Subject: [PrimeNumbers] Re: Primes between squares


            --- In primenumbers@y..., "Jon Perry" <perry@g...> wrote:
            > Is there always a prime between x^2 and (x+n)^2?
            >

            The answer is almost certainly yes, but it's unproven.

            See the last entry on the page:

            http://www.utm.edu/research/primes/notes/conjectures/





            Unsubscribe by an email to: primenumbers-unsubscribe@egroups.com
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          • thomas_ruley
            I was trying to find out about the theorem that there is at least one prime between every pair of consecutive squares. I thought I saw someting about a proof
            Message 5 of 8 , Apr 26 4:33 PM
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              I was trying to find out about the theorem that there is at least one
              prime between every pair of consecutive squares. I thought I saw
              someting about a proof in Archives of Mathematics in 2000, but I
              couldn't find the paper. Does anybody have any information about this
              theorem?

              Thom
            • Décio Luiz Gazzoni Filho
              ... I know it s not the best source, but according to `Proofs from the Book , whose last edition dates to 2000, the problem is still open. Décio [Non-text
              Message 6 of 8 , Apr 27 2:39 AM
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                On Tuesday 26 April 2005 20:33, you wrote:
                > I was trying to find out about the theorem that there is at least one
                > prime between every pair of consecutive squares. I thought I saw
                > someting about a proof in Archives of Mathematics in 2000, but I
                > couldn't find the paper. Does anybody have any information about this
                > theorem?
                >
                > Thom

                I know it's not the best source, but according to `Proofs from the Book',
                whose last edition dates to 2000, the problem is still open.

                Décio


                [Non-text portions of this message have been removed]
              • thomas_ruley
                ... least one ... about this ... Book , ... OK, so it hasn t been proven yet. What about computer testing? I recently wrote a program to test this theorem,
                Message 7 of 8 , Apr 28 5:11 PM
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                  --- In primenumbers@yahoogroups.com, Décio Luiz Gazzoni Filho
                  <decio@d...> wrote:
                  > On Tuesday 26 April 2005 20:33, you wrote:
                  > > I was trying to find out about the theorem that there is at
                  least one
                  > > prime between every pair of consecutive squares. I thought I saw
                  > > someting about a proof in Archives of Mathematics in 2000, but I
                  > > couldn't find the paper. Does anybody have any information
                  about this
                  > > theorem?
                  > >
                  > > Thom
                  >
                  > I know it's not the best source, but according to `Proofs from the
                  Book',
                  > whose last edition dates to 2000, the problem is still open.
                  >
                  > Décio
                  >
                  >
                  > [Non-text portions of this message have been removed]

                  OK, so it hasn't been proven yet. What about computer testing? I
                  recently wrote a program to test this theorem, and was wondering how
                  high I should start testing.

                  Thom
                • Jens Kruse Andersen
                  ... You would have to start above 2*10^17 but I suggest you forget about it. All large prime gaps between smaller primes have been computed:
                  Message 8 of 8 , Apr 28 6:54 PM
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                    thomas_ruley wrote:

                    > OK, so it hasn't been proven yet. What about computer testing? I
                    > recently wrote a program to test this theorem, and was wondering how
                    > high I should start testing.

                    You would have to start above 2*10^17 but I suggest you forget about it.
                    All large prime gaps between smaller primes have been computed:
                    http://www.ieeta.pt/~tos/gaps.html

                    Also see First occurrence prime gaps:
                    http://www.trnicely.net/gaps/gaplist.html#MainTable

                    The largest gap below 2*10^17 is only 1220.
                    Computing a counter example to primes between squares seems highly unlikely
                    to me.

                    --
                    Jens Kruse Andersen
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