I had found this last Wednesday (it was found over the weekend, and
I was sick on Monday/Tuesday). I had been searching for these CPAP5's
with Hans R for over a year now.

I have started documenting the full search and method, along with all
of the stats, but the wife has put me to work around the house
(projects which I put off doing over the winter, yuck!).

The search has been progressing in 2 directions. Hans is using my
CPAPSieve and find_cpapn programs to seach ranges of 2^n+k with
fixed n and variable k. This method allows a much faster PRP test
within PFGW (using the prothlike reduction code). He is searching all
factored candidates when 5 are found in AP with a gap in the range
of 30 to 1500. I had started to use this method also (serveral months
worth about 15 months ago, but numbers of this type are VERY sparse
in candates which might be CPAP-5. They are there, it simply takes a
lot of sieving with a lot of memory on the PC's.

For the search I later settled on, I used primorial-likes of the form
k*n#+b,+0,+30,+60,+90,+120 where b, b+30, b+60, b+90, b+120 have no
factors >= n, and thus all 5 "behave" like primorials. The k is the
variable part of the expression. I used APSieve to trial factor and
with just a couple of big memory fast PC's, I could keep all of my
PRP machines stocked with candidates. I some problems early on with
APSieve; it would exit at the touch of the keyboard, it did not allow
restarting, and it did not save anything until the end. I had
several power outages at work (where one of the sieve boxes is) over
the weekend for about 4 weeks in a row, and bumped the keyboard
serveral times. I contacted Michael Bell (creator of APSieve) to
get the code to enhance it to auto save, and to restart. In working
with it, I found that APSieve had a serious flaw of having round off
problems at just over 2^35 bits. I switched out the 48 (35) bit
math routines and put in my own 62 bit version, and also doubled the
speed in the process. I have put a lot of work into APSieve over the
last 6 months, and now it is actually the fastest publicly available
primorial trial factoring software available. In factoring, I have
about 8 months solid of Athlon 750 (with 448mb) and 8 months solid
on a PIII 650 with 384mb. This translates into about 2.45 PII-400
CPU years. The PRP testing is 8 solid months of the equivilant
of 25 PI-400's (Some Duron's a handful of PII/III 350 to 650's, and
a whole bunch of PMMX 180 to 233's.) Total PRP time (estimate) is
16.6 PII-400 CPU years. All PRP's done with WinPFGW. Proving took
a trivial 10 hours of Duron 750 time.

Before this time, there was only one published known titanic CPAP-4.
During this search, I found 123 AP-4's (121 were CPAP4). I am still
cleaning up the sieved ranges, and expect 0 to 3 more. Hans has 68
AP-4's with 45 not consecutive and 23 consecutive. This puts things
a little in perspective. Also note that the current record for prime
5tuple is only 450 digits long 14519751105*1050#+1042090781+0,2,6,8,12
and it uses the "same" method of search (the primorial + a magic set
of values). A 5-tuple and a CPAP-5 are about identical in difficulty
of finding.

Again, I will be writting up a full document, along with statistics
and a list of all CPrpAP-4's found (by me) shortly.

Jim.

--- In primenumbers@y..., "djbroadhurst" <d.broadhurst@o...> wrote:
> A big hand to Jim and Marcel for the CPAP5:
>
> 9999 142661157626.2411#+71427877 1038 x23 02 Consecutive primes
> arithmetic progression (5,d=30) #0205
> 9999 142661157626.2411#+71427847 1038 x23 02 Consecutive primes
> arithmetic progression (4,d=30) #0205
> 9999 142661157626.2411#+71427817 1038 x23 02 Consecutive primes
> arithmetic progression (3,d=30) #0205
> 9999 142661157626.2411#+71427787 1038 x23 02 Consecutive primes
> arithmetic progression (2,d=30) #0205
> 9999 142661157626.2411#+71427757 1038 x23 02 Consecutive primes
> arithmetic progression (1,d=30) #0205
>
> I'm eager to hear from Jim how this huge feat
> was accomplished.

hehe, I got busted posting the find to primepages before the writeup.
Hans also posted a teaser a few days ago, but no one nibbled at the
bait.

> David