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

15671Re: Largest CPAP differences

Expand Messages
  • jim_fougeron
    Dec 1, 2004
      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
    • Show all 6 messages in this topic