## Largest CPAP differences

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• In September Hans Rosenthal reported a prp CPAP-3 with large difference in ... Such records should be listed somewhere... The Largest Known CPAP s has today
Message 1 of 6 , Nov 30, 2004
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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
• 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:
Message 2 of 6 , Dec 1, 2004
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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
• ... How fortunate that the year is divisible by 3#. Don t expect me to wait until 2010 to beat it :-) Congratulations on beating my ... erm ... improving your
Message 3 of 6 , Dec 1, 2004
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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]
> This is 379 decimal digit CPAP-4 with d=2004 (d chosen for the
> year of finding).

How fortunate that the year is divisible by 3#.
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_
>
> 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

These were found during a CPAP-5 search by Hans and Jim.
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
• ... 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
Message 4 of 6 , Dec 3, 2004
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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
• 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
Message 5 of 6 , Dec 30, 2004
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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
• ... Congratulations. 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
Message 6 of 6 , Dec 31, 2004
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Jim Fougeron wrote:

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

Congratulations.
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
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