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Nuutti Kuosa et al.'s Factorial Prime search.

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  • Phil Carmody
    Good projects never die, they simply reserve the right to take a rest every so often. However, I, and others, think the factorial prime seach has rested quite
    Message 1 of 5 , Nov 1, 2005
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      Good projects never die, they simply reserve the right to take
      a rest every so often. However, I, and others, think the factorial
      prime seach has rested quite long enough!

      Apparently I have the most complete mirror of Nuutti's site, and
      the most up-to-date information, but unfortunately some of it is
      contradictory. Does anyone have any information about completed
      ranges that is more up-to-date than the info at:
      http://fatphil.org/Nuutti/
      http://fatphil.org/Nuutti/minus/minus.html

      I have re-written my factorial sieve, it's now much faster (maybe
      somewhere close to the speed of Mark's excellent multisieve), and
      have been running it for a week or so, and will be making acceptably-
      sieved ranges available for reservation in the next few weeks.
      (My p=6G sieving limit last time was pretty pathetic to be honest!)
      I'll keep sieving for months, obviously.

      Anyway, this was just a little bit of an pre-announcement, so that
      those who are coming to the end of projects, or are between projects,
      can perhaps plan what to do with their CPUs in 2006.

      I'll post again when ranges are available for automatic reservation
      and download, which will be after I am fairly sure that I know
      exactly what has been tested and what hasn't.

      Thanks to Thommy for giving me a kick, I've been meaning to do this
      for a long time.

      Phil

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    • ed pegg
      Is the following true? For any number n with less than 20000 digits, if n+1 or n-1 is an easily factorable smooth number, then the primality/non-primality of n
      Message 2 of 5 , Nov 1, 2005
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        Is the following true?

        For any number n with less than 20000 digits, if n+1 or n-1 is
        an easily factorable smooth number, then the primality/non-primality
        of n can be established with certainty.

        If so, what is the primality proof method called?

        Ed Pegg Jr.
      • Jonathan A. Zylstra
        ... The Prime Pages contain all sorts of useful knowledge about prime numbers. For your question, yes, it s true, it can be established with certainty, and
        Message 3 of 5 , Nov 1, 2005
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          ed pegg wrote:

          > Is the following true?
          >
          > For any number n with less than 20000 digits, if n+1 or n-1 is
          > an easily factorable smooth number, then the primality/non-primality
          > of n can be established with certainty.
          >
          > If so, what is the primality proof method called?
          >
          > Ed Pegg Jr.
          >
          >
          >
          >
          > Unsubscribe by an email to: primenumbers-unsubscribe@yahoogroups.com
          > The Prime Pages : http://www.primepages.org/
          >
          The Prime Pages contain all sorts of useful knowledge about prime
          numbers. For your question, yes, it's true, it can be established with
          certainty, and it's not just limited to 20,000 digits. In fact, by
          modifying the tests, you don't have to completely factor 'N-1' or 'N+1'
          ... you just have to factor them 'enough'. See
          http://primes.utm.edu/prove/index.html for more information on the 'N-1'
          tests and the 'N+1' tests. These pages will also give you references to
          find more detailed information.

          Jonathan Zylstra

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          [Non-text portions of this message have been removed]
        • Jens Kruse Andersen
          ... If you want old contents of a URL then try the Internet Archive: http://www.archive.org http://web.archive.org/web/20040922151554/http://powersum.dhis.org/
          Message 4 of 5 , Nov 1, 2005
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            Phil Carmody wrote:

            > Apparently I have the most complete mirror of Nuutti's site, and
            > the most up-to-date information, but unfortunately some of it is
            > contradictory. Does anyone have any information about completed
            > ranges that is more up-to-date than the info at:
            > http://fatphil.org/Nuutti/
            > http://fatphil.org/Nuutti/minus/minus.html

            If you want old contents of a URL then try the Internet Archive:
            http://www.archive.org

            http://web.archive.org/web/20040922151554/http://powersum.dhis.org/

            --
            Jens Kruse Andersen
          • Jens Kruse Andersen
            ... Yes, this and more is true. Any number n can be proven prime/composite easily , in time O(d^2 * log d * log log d) where d = log n, _if_ enough of the
            Message 5 of 5 , Nov 1, 2005
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              Ed Pegg Jr. wrote:

              > Is the following true?
              >
              > For any number n with less than 20000 digits, if n+1 or n-1 is
              > an easily factorable smooth number, then the primality/non-primality
              > of n can be established with certainty.

              Yes, this and more is true. Any number n can be proven prime/composite
              "easily", in time O(d^2 * log d * log log d) where d = log n, _if_ enough of
              the prime factorization of n+1 or n-1 is known.
              This time is much faster than the average time to find a probable prime.
              The known prime factors of n+/-1 don't have to be small, they just have to be
              proven primes.

              > If so, what is the primality proof method called?

              There are different proof methods with different names, depending on the form
              and factorization percentage of n+/-1. The methods are generally called
              "classical tests".
              See the Prime Pages for details: http://primes.utm.edu/prove/prove3.html

              The popular flexible program PrimeForm/GW implements a BLS proof
              (Brillhart-Lehmer-Selfridge).
              PrimeForm/GW can prove any n up to a million or more digits, if the product of
              known factors of n-1 or n+1 is at least n^(1/3), or a little less for combined
              n-1 and n+1 proofs.

              Some BLS "extensions" not implemented in PrimeForm can prove numbers with less
              n+/-1 factorization.
              David Broadhurst's KP (Konyagin-Pomerance) PARI script can handle n^0.3.
              John Renze's CHG (Coppersmith-Howgrave-Graham) PARI script (currently being
              debugged) can handle down to around n^0.27, depending on the size of n, the
              used computer, and the acceptable time.

              It is utterly hopeless to find enough n+/-1 factorization to easily prove
              primality of a random n above 1000 digits.
              If you get "lucky", you can find a large probable prime cofactor of n-1 or
              n+1. You cannot prove primality of that cofactor significantly easier than of
              n itself, so it doesn't help in practice.

              The largest proven prime n without large n+/-1 factorization is the ECPP
              record with 15071 digits: http://primes.utm.edu/top20/page.php?id=27

              The top-20 for a publicly available program is all with the ECPP program Primo
              with record at 7993 digits: http://www.ellipsa.net/primo/top20.html

              --
              Jens Kruse Andersen
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