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Re: [primeform] 32476090*binomial(1531785651*2^10110+18,23)-1

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  • Jens Kruse Andersen
    Announcement to be linked from the e-group bio. 32476090*C(1531785651*2^10110+18,23)-1 is prime. C(x,y) = x!/((x-y)!*y!) is the binomial function. In 2001
    Message 1 of 3 , Jun 19, 2005
      Announcement to be linked from the e-group bio.

      32476090*C(1531785651*2^10110+18,23)-1 is prime.
      C(x,y) = x!/((x-y)!*y!) is the binomial function.

      In 2001 Michael Angel, Dirk Augustin, Paul Jobling's NewPGen,
      Paul Jobling, Yves Gallot's Proth.exe found this CC3 of the 2nd kind:
      p = 1531785651*2^10107+1, 2p-1, 4p-3
      Let N = 8*(p-1) = 1531785651*2^10110.
      The CC3 gives all factors of N, N+2, N+4, N+8.

      Jens Kruse Andersen used GMP-ECM to find 3 further complete
      factorizations with these prp's:
      (N-4)/(2^2*5*7*31237*286235451569*11609891459483469679)
      (N+5)/(47*2087*240631)
      (N+15)/(3*6343*1655652931*1809537707*376766311088897*7992061535285413)

      They were proved prime with Marcel Martin's Primo by Luigi Morelli,
      V. M. Ulyanov and Pierre Cami. The 3 verified certificates are in
      http://hjem.get2net.dk/carlkruse/certif/kc23.zip

      Andersen sieved k*C(N+18,23)-1 to 5*10^11.
      Andersen, Morelli, Ulyanov, Cami and D├ęcio Luiz Gazzoni Filho made
      Fermat prp tests with PrimeForm/GW. Cami found a prp with k=32476090.

      There is a little more than 30% factorization of C(N+18,23).
      This is insufficient for a BLS primality proof by PrimeForm.
      David Broadhurst made a Konyagin-Pomerance proof:
      http://physics.open.ac.uk/~dbroadhu/cert/kc23c.zip

      This was a PrimeForm e-group project:
      http://primes.utm.edu/bios/page.php?id=339

      --
      Jens Kruse Andersen
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