15671Re: Largest CPAP differences
- Dec 1, 2004I 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:
This is 379 decimal digit CPAP-4 with d=2004 (d chosen for the
year of finding).
--- In email@example.com, "Jens Kruse Andersen"
> In September Hans Rosenthal reported a prp CPAP-3 with largedifference in
> http://groups.yahoo.com/group/primeform/message/4700 :that has
> > d=17676
> > is the largest known common difference for any CPAP-k (k > 2)
> > ever been discovered, as far as I know.largest known
> Such records should be listed somewhere...
> The Largest Known CPAP's has today added a record table with the
> CPAP-k difference for each k:on*, no
> Hans daringly continued:
> > I dare to conjecture that within one year from now (5 Sep 2004)
> > one will find a larger value of d for any CPAP-k. Note that theaverage
> > prime gap length for the form 9337#+k by testing k from 1 to840000001
> > is 9717.until now.
> Some people who know me may be surprised this conjecture has lasted
> Maybe Hans was challenging me - or maybe I just have illusionsabout the world
> of prime constellations revolving around me :-)Alm & Jens
> Anyway, I did not work on it until this month.
> New difference records have been set for CPAP-3 to -7 by Torbjörn
> Kruse Andersen.record.
> k=3: p2506 + 21102n, n=0..2, Alm & Andersen
> Hans' 4003-digit prp CPAP-3 is unproven so it was not official
> 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
> composites. The decimal expansions are at the record page.3,5,7).
> 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
> p133 + 420n is the first known CPAP-k with non-minimal differencefor k > 6.
> Only difference 7# = 210 is known for CPAP-8, -9 and -10.listed
> In fact, only one CPAP-10 is known at all. The discoverers get 4
> records for the price of 1: Longest CPAP, largest CPAP-10, smallestCPAP-10,
> largest CPAP-10 difference.guarantee many
> The record for k=4 is currently very easy with an algorithm to
> composites. It was found in the search for k=5.
> Jens Kruse Andersen
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