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Re: Largest CPAP differences

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  • jim_fougeron
    I have a 2004 addition for CPAP-4 (yes, it was very easy to find) 78006074.883#+k*2004+R is prime for k = [0...3] Using the special crafted R of:
    Message 1 of 6 , Dec 1 1:41 PM
      I have a 2004 addition for CPAP-4 (yes, it was very easy to find)

      78006074.883#+k*2004+R is prime for k = [0...3]

      Using the special crafted R of:

      R=2114567933911681280472790383441990479992379628560566796354424357\
      838755083647935493830969086740559072320624021043028003098506771578\
      705193556075446873904579965613995564655278439586807781631424628487\
      465675366956065147973222327951284073363128838151633809148759244135\
      745954589064984106465530727879107072157718030628641179072983310965\
      3930842047229740770849707899029657847082299

      This is 379 decimal digit CPAP-4 with d=2004 (d chosen for the
      year of finding).

      Jim Fougeron.

      --- In primenumbers@yahoogroups.com, "Jens Kruse Andersen"
      <jens.k.a@g...> wrote:
      > In September Hans Rosenthal reported a prp CPAP-3 with large
      difference in
      > http://groups.yahoo.com/group/primeform/message/4700 :
      >
      > > d=17676
      > > is the largest known common difference for any CPAP-k (k > 2)
      that has
      > > ever been discovered, as far as I know.
      >
      > Such records should be listed somewhere...
      > The Largest Known CPAP's has today added a record table with the
      largest known
      > CPAP-k difference for each k:
      > http://hjem.get2net.dk/jka/math/cpap.htm#difference
      >
      > Hans daringly continued:
      > > I dare to conjecture that within one year from now (5 Sep 2004)
      on*, no
      > > one will find a larger value of d for any CPAP-k. Note that the
      average
      > > prime gap length for the form 9337#+k by testing k from 1 to
      840000001
      > > is 9717.
      >
      > Some people who know me may be surprised this conjecture has lasted
      until now.
      > Maybe Hans was challenging me - or maybe I just have illusions
      about the world
      > of prime constellations revolving around me :-)
      > Anyway, I did not work on it until this month.
      >
      > New difference records have been set for CPAP-3 to -7 by Torbjörn
      Alm & Jens
      > Kruse Andersen.
      >
      > k=3: p2506 + 21102n, n=0..2, Alm & Andersen
      >
      > Hans' 4003-digit prp CPAP-3 is unproven so it was not official
      record.
      >
      > k=4: Same as k=5 (a poor record for CPAP-4)
      >
      > k=5: p272 + 1350n, n=0..4, Andersen
      >
      > k=6: p218 + 840n, n=0..5, Alm & Andersen
      >
      > k=7: p133 + 420n, n=0..6, Alm & Andersen
      >
      > pN is an N-digit prime with no simple expression, chosen to
      guarantee many
      > composites. The decimal expansions are at the record page.
      > PrimeForm/GW prp'ed for the CPAP-3, the GMP library for the others.
      > Marcel Martin's Primo proved all primes.
      >
      > The difference in a CPAP-k must be a multiple of k# (except CPAP-3
      3,5,7).
      > p133 + 420n is the first known CPAP-k with non-minimal difference
      for k > 6.
      > Only difference 7# = 210 is known for CPAP-8, -9 and -10.
      > In fact, only one CPAP-10 is known at all. The discoverers get 4
      listed
      > records for the price of 1: Longest CPAP, largest CPAP-10, smallest
      CPAP-10,
      > largest CPAP-10 difference.
      >
      > The record for k=4 is currently very easy with an algorithm to
      guarantee many
      > composites. It was found in the search for k=5.
      >
      > --
      > Jens Kruse Andersen
    • Jens Kruse Andersen
      ... How fortunate that the year is divisible by 3#. Don t expect me to wait until 2010 to beat it :-) Congratulations on beating my ... erm ... improving your
      Message 2 of 6 , Dec 1 4:31 PM
        Jim Fougeron wrote:

        > I have a 2004 addition for CPAP-4 (yes, it was very easy to find)
        >
        > 78006074.883#+k*2004+R is prime for k = [0...3]
        > This is 379 decimal digit CPAP-4 with d=2004 (d chosen for the
        > year of finding).

        How fortunate that the year is divisible by 3#.
        Don't expect me to wait until 2010 to beat it :-)

        Congratulations on beating my ... erm ... improving your own record!
        Earlier Wednesday Hans mailed me:

        > I searched my databases and found within _less than a minute_
        >
        > k Prime Diff n's DD Year Discoverer(s)/Prover(s)
        > 4 2^3322+712415559243+1440n n=0..3 1001 2002 Rosenthal, Fougeron, Primo
        > 4 2^3320+1308319536235+1470n n=0..3 1000 2002 Rosenthal, Fougeron, Primo

        These were found during a CPAP-5 search by Hans and Jim.
        I had already updated http://hjem.get2net.dk/jka/math/cpap.htm#difference
        It has been updated again.

        The CPAP page keeps me busy. Now it also generates spam!
        See the amazing CPAP Hose Sock at http://hjem.get2net.dk/jka/math/meaning.htm
        I got the mail Tuesday.

        --
        Jens Kruse Andersen
      • Jens Kruse Andersen
        ... I also chose to search an easy improvement with a special d: 46313478 * 1201#/1302643 + x498 + 2310n, for n = 0..3 Using the special crafted x498 at the
        Message 3 of 6 , Dec 3 2:25 PM
          Jim Fougeron wrote:

          > I have a 2004 addition for CPAP-4 (yes, it was very easy to find)
          >
          > 78006074.883#+k*2004+R is prime for k = [0...3]
          >
          > Using the special crafted R of:
          >
          > R=211456......
          >
          > This is 379 decimal digit CPAP-4 with d=2004 (d chosen for the
          > year of finding).

          I also chose to search an easy improvement with a "special" d:

          46313478 * 1201#/1302643 + x498 + 2310n, for n = 0..3

          Using the special crafted x498 at the site.

          d = 2310 = 11# is the smallest possible d for a CPAP-11.
          Only 7 more primes and a few trillion GHz years to go!
          (Yes, I checked the CPAP-4 doesn't extend)

          PrimeForm/GW prp'ed and Marcel Martin's Primo proved.
          This record is going fast. I added a record history to keep up:
          http://hjem.get2net.dk/jka/math/cpap.htm#difference_history

          --
          Jens Kruse Andersen
        • jim_fougeron
          I have followed Jens lead, and on Christmas day, discovered a CPAP-5 with a gap of 2310. This number is: 9400734826*1499#+x632+2310n with n from 0 to 4 and
          Message 4 of 6 , Dec 30 9:29 PM
            I have followed Jens lead, and on Christmas day, discovered a
            CPAP-5 with a gap of 2310. This number is:

            9400734826*1499#+x632+2310n

            with n from 0 to 4 and

            x632 = 29453397765450271545399188085266
            55368252378620585099496385650600431498269661083903
            16211042912735310776015757228962737061429256177227
            59452435429488328389328281466289664367352954006161
            79207095576921259775792175026579617936878099659414
            32837668975308693630297479962123616982055909919099
            17025496933775418577095695897932136276184735064982
            12549583475520940170609152997656163627242028405951
            20483292467767923352271563327090757509953181908766
            84457108535835673100713235902439791043089273743933
            82074800677693506138530042890392327727580262290580
            36426781990814418117965800120148977404531919575260
            38333320588240996195703518136355252551601080488639

            Now, we need a CPAP-6 :) Any takers?

            Jim.


            --- In primenumbers@yahoogroups.com, "Jens Kruse Andersen"
            <jens.k.a@g...> wrote:
            > Jim Fougeron wrote:
            >
            > > I have a 2004 addition for CPAP-4 (yes, it was very easy to find)
            > >
            > > 78006074.883#+k*2004+R is prime for k = [0...3]
            > >
            > > Using the special crafted R of:
            > >
            > > R=211456......
            > >
            > > This is 379 decimal digit CPAP-4 with d=2004 (d chosen for the
            > > year of finding).
            >
            > I also chose to search an easy improvement with a "special" d:
            >
            > 46313478 * 1201#/1302643 + x498 + 2310n, for n = 0..3
            >
            > Using the special crafted x498 at the site.
            >
            > d = 2310 = 11# is the smallest possible d for a CPAP-11.
            > Only 7 more primes and a few trillion GHz years to go!
            > (Yes, I checked the CPAP-4 doesn't extend)
            >
            > PrimeForm/GW prp'ed and Marcel Martin's Primo proved.
            > This record is going fast. I added a record history to keep up:
            > http://hjem.get2net.dk/jka/math/cpap.htm#difference_history
            >
            > --
            > Jens Kruse Andersen
          • Jens Kruse Andersen
            ... Congratulations. Thanks for your interest in the new record category. http://hjem.get2net.dk/jka/math/cpap.htm#difference is updated. Note that it is also
            Message 5 of 6 , Dec 31 7:03 AM
              Jim Fougeron wrote:

              > I have followed Jens lead, and on Christmas day, discovered a
              > CPAP-5 with a gap of 2310. This number is:
              >
              > 9400734826*1499#+x632+2310n
              >
              > Now, we need a CPAP-6 :) Any takers?

              Congratulations.
              Thanks for your interest in the new record category.
              http://hjem.get2net.dk/jka/math/cpap.htm#difference is updated.
              Note that it is also easily the second largest overall CPAP-5, only beaten by
              Jim himself who has the whole top-10.

              For a CPAP-6 with difference 2310, I think I would search around 560 digits.
              For comparison, the largest known 6 simultaneous primes is a 418-digit CC-6 by
              Dirk Augustin: http://hjem.get2net.dk/jka/math/simultprime.htm#records
              Around 16 AP-6's at 560 digits are expected before hitting a CPAP-6 with my
              program.

              It looks pretty hard. I estimate perhaps 5 GHz years if my deep sieve is
              improved to manage sparse forms better.
              I don't have plans to make such a hard CPAP-6 search.

              --
              Jens Kruse Andersen
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