- In September Hans Rosenthal reported a prp CPAP-3 with large difference in

http://groups.yahoo.com/group/primeform/message/4700 :

> d=17676

Such records should be listed somewhere...

> is the largest known common difference for any CPAP-k (k > 2) that has

> ever been discovered, as far as I know.

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

Some people who know me may be surprised this conjecture has lasted until now.

> 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.

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 - Jim Fougeron wrote:

> I have followed Jens lead, and on Christmas day, discovered a

Congratulations.

> CPAP-5 with a gap of 2310. This number is:

>

> 9400734826*1499#+x632+2310n

>

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

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