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Big PrimoProth prime number

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  • dibo12fr
    Hello, I m glad to announce the discovery of two big (most probably the largest known) primoproth prime numbers. As a reminder, these number can be written
    Message 1 of 5 , Oct 28, 2008
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      Hello,
      I'm glad to announce the discovery of two big (most probably the
      largest known) primoproth prime numbers.
      As a reminder, these number can be written like m#*2n +/- 1, m# being
      the primorial of the prime m.
      155161#*2^2405+1 is prime (67961 digits)
      155167#*2^2405+1 is prime (67966 digits)

      The whole job (PRP test and primality proof) has been done with
      winPFGW (1.29), by keeping the exponant constant and increasing the
      value of m.
      During this search, other number have been proven primes :
      17#*2^2405+1
      947#*2^2405+1
      977#*2^2405+1
      2969#*2^2405+1
      31907#*2^2405+1

      To my knowledge, these 2 records cannot be registered in the Prime
      Pages, but at least, it is shared with all of you.

      Best regards.
      Didier Boivin
    • Jens Kruse Andersen
      ... A Proth prime is k*2^n+1 with k
      Message 2 of 5 , Oct 28, 2008
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        Didier Boivin wrote:
        > I'm glad to announce the discovery of two big (most probably the
        > largest known) primoproth prime numbers.
        > As a reminder, these number can be written like m#*2n +/- 1, m# being
        > the primorial of the prime m.
        > 155161#*2^2405+1 is prime (67961 digits)
        > 155167#*2^2405+1 is prime (67966 digits)
        >
        > During this search, other number have been proven primes :
        > 17#*2^2405+1
        > 947#*2^2405+1
        > 977#*2^2405+1
        > 2969#*2^2405+1
        > 31907#*2^2405+1
        >
        > To my knowledge, these 2 records cannot be registered in the Prime
        > Pages, but at least, it is shared with all of you.

        A Proth prime is k*2^n+1 with k < 2^n.
        A PrimoProth prime is based on this, a Proth prime where k is a primorial:
        m#*2^n+1 with m# < 2^n
        However, m#*2^n-1 with m# < 2^n also appears to be allowed.
        See http://www.primenumbers.net/Henri/us/FactPrimus.htm
        The only PrimoProths in your post are those with 17#, 947#, 977#.

        If the smallest primorials are allowed (and I haven't seen a rule excluding
        them) then many of the largest known primes can actually be written as
        PrimoProths, for example all of the forms:
        2^n-1 =1#*2^n-1
        3*2^n+/-1 = 3#*2^(n-1)+/-1
        15*2^n+/-1 = 5#*2^(n-1)+/-1

        If the Prime Pages had a top-20 for PrimoProths without tiny primorials then
        I guess more people would search them and the top-20 limit would be above
        the 5000th prime so they would have qualified for the database anyway.

        --
        Jens Kruse Andersen
      • Jens Kruse Andersen
        ... After looking at more Google hits, it actually appears that the few people searching or listing PrimoProths use different definitions or don t state a
        Message 3 of 5 , Oct 28, 2008
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          I wrote:
          > A Proth prime is k*2^n+1 with k < 2^n.
          > A PrimoProth prime is based on this, a Proth prime where k is a primorial:
          > m#*2^n+1 with m# < 2^n
          > However, m#*2^n-1 with m# < 2^n also appears to be allowed.

          After looking at more Google hits, it actually appears that the few people
          searching or listing PrimoProths use different definitions or don't state a
          definition at all. This confusion may be caused by Henri Lifchitz who
          introduced the concept but himself seems unclear with the definiton and
          inconsistent in the alleged records, both regarding whether there is an
          upper and a lower limit on m#.

          --
          Jens Kruse Andersen
        • Robert
          ... people ... state a ... I was involved some years ago as a coordinator for primes of the series n#/2*2^n+/1, which I called primoproths, influenced by Henri
          Message 4 of 5 , Oct 29, 2008
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            --- In primenumbers@yahoogroups.com, "Jens Kruse Andersen"
            <jens.k.a@...> wrote:
            >
            > I wrote:
            > > A Proth prime is k*2^n+1 with k < 2^n.
            > > A PrimoProth prime is based on this, a Proth prime where k is a
            primorial:
            > > m#*2^n+1 with m# < 2^n
            > > However, m#*2^n-1 with m# < 2^n also appears to be allowed.
            >
            > After looking at more Google hits, it actually appears that the few
            people
            > searching or listing PrimoProths use different definitions or don't
            state a
            > definition at all. This confusion may be caused by Henri Lifchitz who
            > introduced the concept but himself seems unclear with the definiton and
            > inconsistent in the alleged records, both regarding whether there is an
            > upper and a lower limit on m#.
            >
            > --
            > Jens Kruse Andersens
            >

            I was involved some years ago as a coordinator for primes of the
            series n#/2*2^n+/1, which I called primoproths, influenced by Henri
            -see http://home.btclick.com/rwsmith/pp/page1.htm

            As suspected the largest primes discovered involve k=3 and k=15. The
            Riesel series k=15 is being actively searched and is at n=1834000 see:
            http://www.mersenneforum.org/showthread.php?t=6338. k=3 search is well
            known.

            The attractions of primoproths were diminished when it was discovered
            that Payam numbers gave greater density of primes and active searching
            went into abeyance.

            But these are good finds, so congrats
          • dibo12fr
            ... Hello Jens and Robert, Thanks for your valuable comments. I forgot to check the criteria of m#
            Message 5 of 5 , Nov 6, 2008
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              --- In primenumbers@yahoogroups.com, "Jens Kruse Andersen"
              <jens.k.a@...> wrote:
              >

              > A PrimoProth prime is based on this, a Proth prime where k is a
              primorial:
              > m#*2^n+1 with m# < 2^n
              > However, m#*2^n-1 with m# < 2^n also appears to be allowed.
              > See http://www.primenumbers.net/Henri/us/FactPrimus.htm
              > The only PrimoProths in your post are those with 17#, 947#, 977#.
              >

              Hello Jens and Robert,
              Thanks for your valuable comments. I forgot to check the criteria of
              m# < 2^n, mainly because it was a search with constant n.
              Robert points out that these numbers are not records at all.
              Surprinsingly, the site has not been updated since 2002. Maybe due to
              lack of time, or no more interest or contributors.

              BR
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