Loading ...
Sorry, an error occurred while loading the content.
 

Factorization of 4 consecutive 8193-digit numbers

Expand Messages
  • tjw99
    Let n = 378149751*2^27186-2 n = 2*(378149751*2^27185-1) (2*prime) n+1 = 378149751*2^27186-1 (prime) n+2 = 378149751*2^27186 = 2^27186*3^2*7*17*353081 n+3 =
    Message 1 of 5 , May 2, 2010
      Let n = 378149751*2^27186-2

      n = 2*(378149751*2^27185-1) (2*prime)
      n+1 = 378149751*2^27186-1 (prime)
      n+2 = 378149751*2^27186 = 2^27186*3^2*7*17*353081
      n+3 = 378149751*2^27186+1 = 5*1019*24923*43651*989239*P8174

      The 8174-digit cofactor of n+3 has been proven prime through a joint effort of Geoffrey Hird and myself, and makes the Top-20 for ECPP proofs:

      http://primes.utm.edu/primes/page.php?id=92563

      This prime is currently the fifth-largest candidate certified by Primo, with a certificate available at:

      http://www.ellipsa.eu/public/primo/files/ecpp8174.zip

      The two primes 378149751*2^27185-1 and 378149751*2^27186-1 are a Sophie Germain byproduct of a recent CC3 (1st kind) search.

      n-1 = 378149751*2^27186-3 = 3*53*127*7043*626921*24117041920337*C8166
      n+4 = 378149751*2^27186+2 = 2*11*561139037952529*C8177

      n-1 and n+4 have been tested with 74 ECM curves at B1=11000.

      Tom
    • Jens Kruse Andersen
      ... Congratulations! http://users.cybercity.dk/~dsl522332/math/consecutive_factorizations.htm is updated with a note that the certificate is being verified.
      Message 2 of 5 , May 3, 2010
        Tom wrote:
        > Let n = 378149751*2^27186-2
        >
        > n = 2*(378149751*2^27185-1) (2*prime)
        > n+1 = 378149751*2^27186-1 (prime)
        > n+2 = 378149751*2^27186 = 2^27186*3^2*7*17*353081
        > n+3 = 378149751*2^27186+1 = 5*1019*24923*43651*989239*P8174
        >
        > The 8174-digit cofactor of n+3 has been proven prime through a joint
        > effort of Geoffrey Hird and myself, and makes the Top-20 for ECPP proofs

        Congratulations!
        http://users.cybercity.dk/~dsl522332/math/consecutive_factorizations.htm
        is updated with a note that the certificate is being verified. Marcel Martin
        lists it on the Primo Top-20 and I assume there will be no problems.
        I'm pleasantly surprised that somebody has run Primo at 8174 digits for
        my record page.

        --
        Jens Kruse Andersen
      • djbroadhurst
        ... When http://primes.utm.edu/primes/page.php?id=92563 was posted, by Geoffrey, I wondered: what cunning problem might this prime solve? Thanks, Tom, for your
        Message 3 of 5 , May 4, 2010
          --- In primeform@yahoogroups.com,
          "tjw99" <tjw99@...> wrote:
          >
          > Let n = 378149751*2^27186-2
          >
          > n = 2*(378149751*2^27185-1) (2*prime)
          > n+1 = 378149751*2^27186-1 (prime)
          > n+2 = 378149751*2^27186 = 2^27186*3^2*7*17*353081
          > n+3 = 378149751*2^27186+1 = 5*1019*24923*43651*989239*P8174

          When
          http://primes.utm.edu/primes/page.php?id=92563
          was posted, by Geoffrey, I wondered:
          what cunning problem might this prime solve?

          Thanks, Tom, for your denouement, now added as a comment.

          David
        • Jens Kruse Andersen
          ... When I saw Tom Wu in the prover code I felt certain about the problem it solved and only wondered about a little detail. A few tests confirmed my
          Message 4 of 5 , May 4, 2010
            David wrote:
            > When
            > http://primes.utm.edu/primes/page.php?id=92563
            > was posted, by Geoffrey, I wondered:
            > what cunning problem might this prime solve?

            When I saw Tom Wu in the prover code I felt certain about the problem it
            solved and only wondered about a little detail.
            A few tests confirmed my assumption about the problem and I posted the below
            before the record had been announced or submitted.

            ----- Original Message -----
            From: "Jens Kruse Andersen" <jens.k.a@...>
            To: "tjw99" <tjw99@...>
            Sent: Monday, May 03, 2010 2:27 AM
            Subject: Proven factorization of 4 consecutive 8193-digit numbers

            > Congratulations on http://primes.utm.edu/primes/page.php?id=92563
            > I'm honored that somebody would run Primo at 8174 digits for my record page.
            >
            > I'm a little curious why it was submitted as
            > (378149751*2^27186+1)/(989239*5542921182935)
            >
            > I would have expected either
            > (378149751*2^27186+1)/5483273808085436465 or
            > (378149751*2^27186+1)/(5*1019*24923*43651*989239)
            > but it's not important for the Prime Pages.
            >
            > --
            > Jens Kruse Andersen

            It turned out the latter form was submitted but the Prime Pages
            canonicalization converted it.

            --
            Jens Kruse Andersen
          • djbroadhurst
            ... thus confirming that Jens is quicker on the uptake than am I. However, my slower brain recently had a (perhaps) neat idea that may (or may not) result in
            Message 5 of 5 , May 4, 2010
              --- In primeform@yahoogroups.com,
              "Jens Kruse Andersen" <jens.k.a@...> wrote:

              > When I saw Tom Wu in the prover code I felt certain about
              > the problem it solved

              thus confirming that Jens is quicker on the uptake than am I.

              However, my slower brain recently had a (perhaps) neat
              idea that may (or may not) result in another update for
              Jens to perform, perhaps not before too long.

              Festina lente!

              David
            Your message has been successfully submitted and would be delivered to recipients shortly.