- Here is a new AP9 record at 401 digits:-

(805227062+54790161*n)*941#+1 is prime for n=0..8

Another slightly smaller AP9 was found a little earlier (see below):-

(434578610+97913159*n)*941#+1 is prime for n=0..8

All confirmed prime with PFGW -tc

Input/output statistics:-

Numbers tested by NewPGen: 1.5*10^9

NewPGen reduced these (by sieving to 1.0T) to: 3.74*10^8

Numbers tested by PFGW: 3.0*10^8

PRP's found by PFGW: 1.6*10^7

AP8's found: 95

AP9's found: 2

Run-time statistics:-

NewPGen: 7 GHz-days

PFGW: 87 GHz-days

Pascal program to find AP's: 11 GHz-days

Remarkably, both AP9's were actually found by manually extending the 14

AP8's whose 9th term was outside the range examined by the AP-detector.

The slightly smaller one was found a few msecs earlier, so was for that

length of time the record holder.

(Does that qualify it for a listing on your nice page, Jens?:-)

The number of AP8's is about as expected to find one AP9 - the second

one must have been popped in by Santa Klaus.

-Mike Oakes - --- In primeform@yahoogroups.com, mikeoakes2@... wrote:
>

Congrats. Also to Ken for an AP7:

> Here is a new AP9 record at 401 digits:-

> (805227062+54790161*n)*941#+1 is prime for n=0..8

http://primes.utm.edu/primes/page.php?id=79092

David

Note to Chris: Ken's will need a space adding to

the comment if it is to get archived as an AP. > Congrats. Also to Ken for an AP7:

Let me echo David, and others, congratulations! Nice job. (I fixed the comments)

>

> http://primes.utm.edu/primes/page.php?id=79092

CC- Mike Oakes wrote:
> Here is a new AP9 record at 401 digits:-

Congratulations on another good AP record!

> (805227062+54790161*n)*941#+1 is prime for n=0..8

>

> Another slightly smaller AP9 was found a little earlier

> (see below):-

> (434578610+97913159*n)*941#+1 is prime for n=0..8

>

> The slightly smaller one was found a few msecs earlier, so was

> for that length of time the record holder.

> (Does that qualify it for a listing on your nice page, Jens?:-)

That put you ahead: Beating my AP9 changed our AP score from 5-5

to 6-4 in your favour. Markus Frind also has 4. And an hour ago

Jaroslaw Wroblewski beat my AP20, so now I have 3 (same as Ken

Davis). I started 2006 with 11 (and was back at 11 six months ago),

but it's nice to see my page is creating attention for subtitanic

AP's.

http://hjem.get2net.dk/jka/math/aprecords.htm is updated (twice).

The two AP9 discoveries were so close that I only list the largest.

My page only allows proven primes, so if I did consider both AP9 then

the primality proofs would determine it (assuming the PRP AP's were

found before the proofs). But should computer testing order, or human

reading order of computer output, be decisive if they are different?

Chris chose reading order at http://primes.utm.edu/notes/by_year.html

It's also discussed at

http://www.mersenneforum.org/showthread.php?p=68387

Ken Davis has now found four AP7 for k*3011#+1. The latest doesn't

have the largest prime, so my page doesn't list it.

The recent AP records by Jaroslaw Wroblewski have an interesting form:

AP20: 9372688136871853 + 43#*n, n = 0..19

AP20: 11735227242889999 + 43#*n, n = 0..19

AP18: 398182070391807316627 + 53#*n, n = 0..17

He searched long AP's with exactly those progressions, so the

algorithm was "search simultaneous primes with fixed pattern" and not

the usual "compute a prime pool and test it for AP's".

The used primorial 43# is too large to qualify his two AP20 as the

first allowed cases of more than 18 simultaneous primes at

http://hjem.get2net.dk/jka/math/simultprime.htm

(His AP submissions didn't mention this page or simultaneous primes)

A reason to disallow AP's with small primes and so large primorial

progression: The smallest "simultaneous" cases become far smaller

than tuplets, Cunningham chains and most other forms (smallest 18-

tuplet above 100 has 25 digits). Any number p without a prime factor

<= 43 is an admissible start for an AP20 (or an AP43) with

progression 43#, since p + 43#*n hever has a factor <= 43 if p

doesn't.

"20 consecutive numbers without a prime factor <= 43 are all prime"

should clearly not be allowed as 20 "simultaneous" primes (why stop

at 43 if this was allowed?). But it's a judgement call when to allow

an AP which avoids lots of small factors in a more "natural" way. The

rules have not changed since the page was created, but they are not

set in stone.

--

Jens Kruse Andersen - Jens Kruse Andersen wrote:-

>And an hour ago Jaroslaw Wroblewski beat my AP20,

But you have a terrifically good one: that AP15: nearly 2 years it has

>so now I have 3 (same as Ken Davis).

>

stood, and I'm not about to take it on.

Quality is an important as quantity!

And it was your milestone post

http://tech.groups.yahoo.com/group/primenumbers/message/15965

that initiated my own (and I guess several others') interest in the

subject.

Best regards

Mike