## 15671Re: Largest CPAP differences

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• 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
> 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
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