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Re: [PrimeNumbers] Re: Carol/Kynea new records

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  • Cletus Emmanuel
    Predrag, Thank you. (Sorry if this seems off-topic and seems like poetry - it probably is) Carol G. Kirnon is my best female friend in the whole wide world.
    Message 1 of 20 , Feb 23, 2004
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      Predrag,
      Thank you.
      (Sorry if this seems off-topic and seems like poetry - it probably is) Carol G. Kirnon is my best female friend in the whole wide world. She was the first girl to steal my heart when we were in high school. Therefore, since math is my love and she is my love, I named the first set of numbers after her. The second set, and really the second set, because I encountered them days after is named for the baby girl that had the greatest inpact on my life so far, Kynea R. Griffith (she is ten years old now). I hope some day when people talk about Carol and Kynea numbers, they will know a little bit about the two. Hopefully soon, I will include a biography of the two on Steven Harvey's site.

      -----Cletus Emmanuel

      pminovic <pminovic@...> wrote:
      Cletus,
      Sorry, I was not aware of your GC, but I found it now at the bottom
      of Steven Harvey's CK page. By the way, the comment "Generalized
      Carol" was attached to Ballinger's base 6 GC when I submitted my
      prime. At the time it was already confirmed to be prime. I guess the
      editor of the site removed the comment manually since he doesn't
      approve such "unofficial" comments (?).

      Bt the way, who are Carol and Kynea and why are these forms named
      after them?

      And congrats on your large PRP, (2^148330+1)^4-2, the largest
      reported so far.

      Regards,
      Predrag


      --- In primenumbers@yahoogroups.com, Cletus Emmanuel <cemmanu@y...>
      wrote:
      > Congrats to the two of you on your findings. I went to the Top-
      5000 and did not see the comments next to either number....
      > There is Generalized Carol Number in the form ---> GC = (273*2^k -
      1)^2 - 2. This form is more dense than the regular Carol Number. GC
      is prime for k=100935 with 60774 digits, 607 digits more than Ray's.




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    • Cletus Emmanuel
      Congrats on your new Kynea find... -- 2/23/4 Minovic finds (2^102435+1)^2-2, 61673 digits. That should be Kynea 43 and the previous one listed below is Kynea
      Message 2 of 20 , Feb 23, 2004
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        Congrats on your new Kynea find...
        -->2/23/4 Minovic finds (2^102435+1)^2-2, 61673 digits.

        That should be Kynea 43 and the previous one listed below is Kynea 44.

        pminovic <pminovic@...> wrote:
        Congrats to Ray Ballinger who just reported the largest
        GenCarol prime so far:
        (6^38660-1)^2-2 60167 digits!
        Yesterday I found the new largest Kynea prime:
        (2^110615+1)^2-2 66597 digits!
        Ray's prime has the "Generalized Carol" comment on the Top-5000
        status page, but yesterday after I submitted mine with
        the "Kynea" comment, the comment disappeared when I checked
        again a few hours later. No idea why (?)

        BTW, I'd like to thank Thomas Wolter who sieved Kynea's in the
        100-150k range to 63 bn, Steven Harvey who kindly passed me
        the sieving results, and Mark Rodenkirch for the new version
        of MultiSieve which is now indeed very fast! I sieved to 314 bn
        reducing the number of candidates in the 102-130 range from
        2882 to 2700. And I was lucky to begin testing from n=110,000 on
        a fast machine which produced the prime in several hours!
        I'm going to complete the 102-130 range (Kynea only) but
        Kynea 130-150k and Carol 101-150k (with a gap around 110k)
        ranges are available since Thomas cancelled his reservation.
        Note that the forms are very dense, almost 100 candidates per
        1000 n's after substential sieving.

        Predrag





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      • pminovic
        Cletus, Thank you. You won t beleive it but I just ran into yet another Kynea prime: (2^108888+1)^2-2, 65558 digits. The range appears to be populated with
        Message 3 of 20 , Feb 24, 2004
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          Cletus,
          Thank you. You won't beleive it but I just ran into yet another Kynea
          prime:
          (2^108888+1)^2-2, 65558 digits.
          The range appears to be populated with primes!
          Thank you too for explanation who Carol and Kynea are.
          I wish all the best to all of you!

          Predrag


          --- In primenumbers@yahoogroups.com, Cletus Emmanuel <cemmanu@y...>
          wrote:
          > Congrats on your new Kynea find...
          > -->2/23/4 Minovic finds (2^102435+1)^2-2, 61673 digits.
          >
          > That should be Kynea 43 and the previous one listed below is Kynea
          44.
        • Cletus Emmanuel
          Pregrag, Great! With every find, I feel good inside. I haven t found a Carol/Kynea prime since 2002, and yet I feel connected to every find. That is because
          Message 4 of 20 , Feb 24, 2004
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            Pregrag,
            Great! With every find, I feel good inside. I haven't found a Carol/Kynea prime since 2002, and yet I feel connected to every find. That is because every find motivates me to continue working on my conjecture, which I know in time I would be able to prove, thus making finding Carol/Kynea primes more simple. I do hope that one of you can take on the challenge of searching for a 1M digit Carol/Kynea or even one that would break the current prime record.

            P.S.: Now I am officially of the top-ten list of largest Carol/Kynea Primes, yet I smile gracefully. Thank you all for taking an interest in what I consider my favorite of many hobbies (searching for Carol/Kynea Primes). Thanks especially to Steven Harvey and Rob Binnekamp for helping search the first 100K numbers.

            By the way Steven, the first Kynea prime is (2^0+1)^2 - 2 = 2 and is the only exception to the "Emmanuel Conjecture" because K(0) is even.

            pminovic <pminovic@...> wrote:
            Cletus,
            Thank you. You won't beleive it but I just ran into yet another Kynea
            prime:
            (2^108888+1)^2-2, 65558 digits.
            The range appears to be populated with primes!
            Thank you too for explanation who Carol and Kynea are.
            I wish all the best to all of you!

            Predrag


            --- In primenumbers@yahoogroups.com, Cletus Emmanuel <cemmanu@y...>
            wrote:
            > Congrats on your new Kynea find...
            > -->2/23/4 Minovic finds (2^102435+1)^2-2, 61673 digits.
            >
            > That should be Kynea 43 and the previous one listed below is Kynea
            44.




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          • Zak Seidov
            Dear prime gurus, is this Mathematica s prime really prime: 17911135064090123664377811162569837 (Sloan s A092540), thanks, Zak
            Message 5 of 20 , Feb 27, 2004
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              Dear prime gurus,
              is this Mathematica's prime really prime:
              17911135064090123664377811162569837 (Sloan's A092540),
              thanks,
              Zak
            • Décio Luiz Gazzoni Filho
              ... Hash: SHA1 ... Yes, PARI/GP says so. Décio ... Version: GnuPG v1.2.3 (GNU/Linux) iD8DBQFAP2zmFXvAfvngkOIRAosFAJ921iJammIUgcu5uNN4WR3dBxNa3ACfT/as
              Message 6 of 20 , Feb 27, 2004
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                -----BEGIN PGP SIGNED MESSAGE-----
                Hash: SHA1

                On Friday 27 February 2004 12:52, you wrote:
                > Dear prime gurus,
                > is this Mathematica's prime really prime:
                > 17911135064090123664377811162569837 (Sloan's A092540),

                Yes, PARI/GP says so.

                Décio
                -----BEGIN PGP SIGNATURE-----
                Version: GnuPG v1.2.3 (GNU/Linux)

                iD8DBQFAP2zmFXvAfvngkOIRAosFAJ921iJammIUgcu5uNN4WR3dBxNa3ACfT/as
                GQYoDBxlfMRfUiX4hw5SwWE=
                =cgfu
                -----END PGP SIGNATURE-----
              • Zak Seidov
                Thanks, Filho! Zak
                Message 7 of 20 , Feb 27, 2004
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                  Thanks, Filho!
                  Zak
                  --- In primenumbers@yahoogroups.com, Décio Luiz Gazzoni Filho
                  <decio@r...> wrote:
                  > -----BEGIN PGP SIGNED MESSAGE-----
                  > Hash: SHA1
                  >
                  > On Friday 27 February 2004 12:52, you wrote:
                  > > Dear prime gurus,
                  > > is this Mathematica's prime really prime:
                  > > 17911135064090123664377811162569837 (Sloan's A092540),
                  >
                  > Yes, PARI/GP says so.
                  >
                  > Décio
                  > -----BEGIN PGP SIGNATURE-----
                  > Version: GnuPG v1.2.3 (GNU/Linux)
                  >
                  > iD8DBQFAP2zmFXvAfvngkOIRAosFAJ921iJammIUgcu5uNN4WR3dBxNa3ACfT/as
                  > GQYoDBxlfMRfUiX4hw5SwWE=
                  > =cgfu
                  > -----END PGP SIGNATURE-----
                • Alan Eliasen
                  When you call this a Mathematica s prime, does that mean that this is a prime you ve tested using Mathematica s (probabilistic) PrimeQ method? If so, the
                  Message 8 of 20 , Feb 27, 2004
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                    When you call this a "Mathematica's prime," does that mean that this is a
                    prime you've tested using Mathematica's (probabilistic) PrimeQ method? If so,
                    the probability that Mathematica returned a wrong answer is much less than,
                    say, the probability that there was a TCP/IP or hard-disk error in
                    transmission that corrupted the original number you sent in your e-mail.

                    If you want Mathematica to prove its primality, you can do that too:

                    <<NumberTheory`PrimeQ`
                    ProvablePrimeQ[17911135064090123664377811162569837]
                    PrimeQCertificate[17911135064090123664377811162569837]

                    There's more information on Mathematica's primality proving routines here:

                    http://documents.wolfram.com/v5/Add-onsLinks/StandardPackages/NumberTheory/PrimeQ.html

                    --
                    Alan Eliasen | "You cannot reason a person out of a
                    eliasen@... | position he did not reason himself
                    http://futureboy.homeip.net/ | into in the first place."
                    | --Jonathan Swift
                  • Zak Seidov
                    But PrimeQ works for n
                    Message 9 of 20 , Feb 27, 2004
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                      But PrimeQ works for n < 10^16 (?), Zak
                      --- In primenumbers@yahoogroups.com, Alan Eliasen <eliasen@m...>
                      wrote:
                      >
                      > When you call this a "Mathematica's prime," does that mean that
                      this is a
                      > prime you've tested using Mathematica's (probabilistic) PrimeQ
                      method? If so,
                      > the probability that Mathematica returned a wrong answer is much
                      less than,
                      > say, the probability that there was a TCP/IP or hard-disk error in
                      > transmission that corrupted the original number you sent in your e-
                      mail.
                      >
                      > If you want Mathematica to prove its primality, you can do that
                      too:
                      >
                      > <<NumberTheory`PrimeQ`
                      > ProvablePrimeQ[17911135064090123664377811162569837]
                      > PrimeQCertificate[17911135064090123664377811162569837]
                      >
                      > There's more information on Mathematica's primality proving
                      routines here:
                      >
                      > http://documents.wolfram.com/v5/Add-
                      onsLinks/StandardPackages/NumberTheory/PrimeQ.html
                      >
                      > --
                      > Alan Eliasen | "You cannot reason a person out of
                      a
                      > eliasen@m... | position he did not reason himself
                      > http://futureboy.homeip.net/ | into in the first place."
                      > | --Jonathan Swift
                    • Alan Eliasen
                      ... I think you may have misinterpreted a passage in the documentation. It notes the following: * PrimeQ first tests for divisibility using small primes, then
                      Message 10 of 20 , Feb 27, 2004
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                        Zak Seidov wrote:
                        > But PrimeQ works for n < 10^16 (?), Zak

                        I think you may have misinterpreted a passage in the documentation. It
                        notes the following:

                        * PrimeQ first tests for divisibility using small primes, then uses the
                        Miller-Rabin strong pseudoprime test base 2 and base 3, and then uses a Lucas
                        test.

                        * As of 1997, this procedure is known to be correct only for n < 10^16, and
                        it is conceivable that for larger it could claim a composite number to be prime.

                        This *doesn't* mean that PrimeQ doesn't work for numbers greater than
                        10^16, only that this test has been shown to be sufficient to *prove*
                        primality for all numbers less than 10^16. (The bound is probably higher
                        now... does anyone know?) Beyond that, it's not *proven* to always return the
                        right answer, but the probability is very high indeed that it will. The
                        probability of your hardware failing and returning the wrong answer is much,
                        much higher. In fact, there are *no* known counter-examples, as far as I know.

                        If you're familiar with the Rabin-Miller test, it's a probabilistic test.
                        It may incorrectly declare a composite number to be prime, but the probability
                        of error decreases with the number of bases that are tested. The probability
                        of error decreases quite rapidly as the numbers become large. See:

                        Damgård, I.; Landrock, P.; and Pomerance, C. "Average Case Error Estimates for
                        the Strong Probable Prime Test." Math. Comput. 61, 177-194, 1993.

                        However, this is then combined with a Lucas test. The combination of these
                        tests is often called the "Baillie-PSW" test.

                        Someone may correct me on this, but there are *no known* composite numbers
                        that are declared prime by this combination of tests, but it hasn't been
                        proven that there *can't be.* Carl Pomerance speculated that there might be
                        some, and gave hints on how said numbers could be constructed:

                        http://www.pseudoprime.com/dopo.pdf

                        It's a *very* strong test. Marcel Martin (author of the excellent PRIMO
                        prime-proving program) conjectured last month that "it is presumable that no
                        composite less than, say, 10000 digits can fool this test." See here:

                        http://tinyurl.com/3go6h

                        Several people have offers of money if you can find a counter-example. Not
                        a huge amount, but the combined totals will be easily over $1000.

                        --
                        Alan Eliasen | "You cannot reason a person out of a
                        eliasen@... | position he did not reason himself
                        http://futureboy.homeip.net/ | into in the first place."
                        | --Jonathan Swift



                        --
                        Alan Eliasen | "You cannot reason a person out of a
                        eliasen@... | position he did not reason himself
                        http://futureboy.homeip.net/ | into in the first place."
                        | --Jonathan Swift
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