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R109297

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  • julienbenney
    It s odd I just came to think of primes as I came in after the first rain in Melbourne for weeks. I was reminded of the proof of the primality of R317 and just
    Message 1 of 6 , Apr 21, 2007
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      It's odd I just came to think of primes as I came in after the first rain in Melbourne for weeks.
      I was reminded of the proof of the primality of R317 and just looked up "repunit prime" on
      Wikipedia, expecting to find nothing at all that I did not already know.

      When I checked it, however, I found, to my greatest surprise, that Harvey Dubner had found
      yet another probable repunit prime with 109297 copies of the number one. It was Dubner
      who found, after a lengthy search when for a long time he did not expect to find a repunit
      prime, the previous probable prime repunits R49081 and R86453.

      The distance between R1031 and R49081 and the fact that there seemed to be three "pairs"
      of repunit primes or PRPs (19 and 23, 317 and 1031, 49081 and 86453) made me think
      intuitively that the next PRP repunit would be extremely large. I imagined when I first saw
      R86453 to be a probable prime that the next PRP repunit would have over one million digits -
      sometimes imagining it to be one of the numbers of the prime quadruples beginning with
      1,006,301 and 1,006,331. Having seen R109297, I can imagine the next PRP repunit having
      several million digits - Dubner himself intends only going to 200,000 and I do not know what
      to say if there is another PRP repunit up to that small a number of digits.
    • Jens Kruse Andersen
      ... Glad to be of service. (I saw Dubner s announcement and added it to Wikipedia a few hours later) ... Actually, Dubner found R49081 and Lew Baxter found
      Message 2 of 6 , Apr 21, 2007
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        julienbenney wrote:
        > I was reminded of the proof of the primality of R317 and just looked up
        > "repunit prime" on Wikipedia, expecting to find nothing at all that I did
        > not already know.
        >
        > When I checked it, however, I found, to my greatest surprise, that Harvey
        > Dubner had found yet another probable repunit prime with 109297 copies
        > of the number one.

        Glad to be of service. (I saw Dubner's announcement and added it to
        Wikipedia a few hours later)

        > It was Dubner who found, after a lengthy search when for a long time
        > he did not expect to find a repunit prime, the previous probable
        > prime repunits R49081 and R86453.

        Actually, Dubner found R49081 and Lew Baxter found R86453:
        http://listserv.nodak.edu/cgi-bin/wa.exe?A2=ind0704&L=nmbrthry&T=0&P=178

        > Dubner himself intends only going to 200,000 and I do not know what
        > to say if there is another PRP repunit up to that small a number of
        > digits.

        Dubner has already reached 200000 with no new PRP:
        http://tech.groups.yahoo.com/group/primeform/message/8546

        --
        Jens Kruse Andersen
      • julienbenney
        ... Well, I half expected that. All I could read on my browser was that it would take more memory to search from 200000 to 300000 than to search up to 200000.
        Message 3 of 6 , Apr 22, 2007
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          > Dubner has already reached 200000 with no new PRP:
          > http://tech.groups.yahoo.com/group/primeform/message/8546

          Well, I half expected that. All I could read on my browser was that it would take more
          memory to search from 200000 to 300000 than to search up to 200000. It seems to me as if
          new programs are needed to find another PRP repunit. I always imagine the next one to be in
          the millions of 1s, not below 300000.

          Maybe they should search other bases. I recall reading tables of generalised repunit primes
          for bases up to 51 (for base 51, there were none at all up to the limit of n=2609). For
          instance, what is the next number in the sequence 3, 43, 73, 487, 2579, 8741, ...??
        • Phil Carmody
          ... The exponents will grow on average exponentially. It looks like a faster growth than Mersennes, and it should be possible to perform the same kind of
          Message 4 of 6 , Apr 22, 2007
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            --- julienbenney <jpbenney@...> wrote:
            > > Dubner has already reached 200000 with no new PRP:
            > > http://tech.groups.yahoo.com/group/primeform/message/8546
            >
            > Well, I half expected that. All I could read on my browser was that it would
            > take more
            > memory to search from 200000 to 300000 than to search up to 200000. It seems
            > to me as if
            > new programs are needed to find another PRP repunit. I always imagine the
            > next one to be in
            > the millions of 1s, not below 300000.

            The exponents will grow on average exponentially. It looks like a faster growth
            than Mersennes, and it should be possible to perform the same kind of density
            analysis that has been done for Mersennes. Millions certainly isn't
            unnecessarily pessimistic.

            Phil

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          • Mark Underwood
            ... rain in Melbourne for weeks. Glad to see your part of Australia is finally getting some rain. It s pretty dry down there I hear. I just noticed something
            Message 5 of 6 , Apr 23, 2007
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              --- In primenumbers@yahoogroups.com, "julienbenney" <jpbenney@...> wrote:
              >
              > It's odd I just came to think of primes as I came in after the first
              rain in Melbourne for weeks.

              Glad to see your part of Australia is finally getting some rain. It's
              pretty dry down there I hear.


              I just noticed something in common with all the prime and probable
              prime repunits: R(2), R(19), R(23), R(317), R(1031), R(49081),
              R(86453) and R(109297).

              They are all prime of course, but all of their values mod (10^x) are
              also a prime, a power of 3 times a prime, or one.

              For instance take the largest prp prime repunit R(109297):

              109297 is prime
              109297 mod 100000 = 9297 = 3^2 * 1033
              109297 mod 10000 = 9297 = 3^2 * 1033
              109297 mod 1000 = 297 = 3^3 * 11
              109297 mod 100 = 97
              109297 mod 10 = 7

              Furthermore, notice that with the derived 1033 prime factor above, it
              too has all its values mod 10^x as prime, a power of 3 times a prime,
              or one.

              And so on with the rest of the prime repunits. Coincidence? I doubt
              it would hold for all larger prime repunits, but it's fun observing
              what is likely the law of small numbers in effect.

              Mark
            • Andy Steward
              From: julienbenney Sent: 22 April 2007 14:32 ... Phi(4229,51) is PrP. The first base for which I have not yet found a PrP is 152. There are 34 others under
              Message 6 of 6 , Apr 25, 2007
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                From: julienbenney
                Sent: 22 April 2007 14:32

                >Maybe they should search other bases. I recall reading tables of
                >generalised repunit primes for bases up to 51 (for base 51, there were
                >none at all up to the limit of n=2609).

                Phi(4229,51) is PrP. The first base for which I have not yet found a PrP
                is 152. There are 34 others under 1000 (excepting perfect powers)
                without a PrP, all have been searched for (prime) exponents up to 2^15.

                I can find no hints as to covering sets that would exclude any of these.

                >For instance, what is the next number in the sequence 3, 43, 73, 487,
                >2579, 8741, ...??

                37441 is a candidate. The answer lies in (10177,37441].
                (Base-15 PrP GRUs.)

                I should do a OEIS search and update whichever sequences I can. Last
                time I checked, I had "first base for each exponent" and "first exponent
                for each base" considerably improved from the published sequences, but
                that was years ago and someone else may have been working on them.

                There's too much "real life" going on atm ;-)

                Best Regards,
                Andy Steward
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