- 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 :

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

> 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)

> > 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.

until now.

>

> Some people who know me may be surprised this conjecture has lasted

> Maybe Hans was challenging me - or maybe I just have illusions

about 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

>

guarantee many

> 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 difference

for 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, smallest

CPAP-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 - Jim Fougeron wrote:

> I have a 2004 addition for CPAP-4 (yes, it was very easy to find)

How fortunate that the year is divisible by 3#.

>

> 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).

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_

These were found during a CPAP-5 search by Hans and Jim.

>

> 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

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 - Jim Fougeron wrote:

> I have a 2004 addition for CPAP-4 (yes, it was very easy to find)

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

>

> 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).

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 - 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 - 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