- Hi All,

Having started to thing Poisson had it in for me (yet again) as I'd found 107 (mostly central AP9's giving 205 chances) without an Ap10, I've hit the Jackpot.

originally detected as an AP8 of the form

(3434538441+n*61959394)*653#+1 n= 0-6

when extended downwards turned out to be not only an AP10 but also an AP11 giving me

two records at once

The following descibes a AP11 with 282-283 digits

(3186700865+n*61959394)*653#+1 n= 0-10

Numbers tested by NewPGen: 3*10^9

seived by NewPGen

PRP's found by PFGW: 53,759,445 (1 in every 55.8)

AP9's found: 107

AP10's found: 2

AP11's found: 1

Cheers

Ken

p.s. the only downside is that, as I was confident that an ap10 was close to being found,

I'd already started preparation for an Ap11 search and had already done the newpgenning

80% of the pfgwing in preparation.

Primality testing (3186700865+0*61959394)*653#+1 [N-1, Brillhart-Lehmer-Selfridge]

Running N-1 test using base 2

Calling Brillhart-Lehmer-Selfridge with factored part 34.19%

(3186700865+0*61959394)*653#+1 is prime! (0.0365s+0.0014s)

Primality testing (3186700865+1*61959394)*653#+1 [N-1, Brillhart-Lehmer-Selfridge]

Running N-1 test using base 3

Calling Brillhart-Lehmer-Selfridge with factored part 33.55%

(3186700865+1*61959394)*653#+1 is prime! (0.0235s+0.0015s)

Primality testing (3186700865+2*61959394)*653#+1 [N-1, Brillhart-Lehmer-Selfridge]

Running N-1 test using base 7

Calling Brillhart-Lehmer-Selfridge with factored part 33.87%

(3186700865+2*61959394)*653#+1 is prime! (0.0238s+0.0014s)

Primality testing (3186700865+3*61959394)*653#+1 [N-1, Brillhart-Lehmer-Selfridge]

Running N-1 test using base 11

Calling Brillhart-Lehmer-Selfridge with factored part 33.97%

(3186700865+3*61959394)*653#+1 is prime! (0.0233s+0.0014s)

Primality testing (3186700865+4*61959394)*653#+1 [N-1, Brillhart-Lehmer-Selfridge]

Running N-1 test using base 19

Calling Brillhart-Lehmer-Selfridge with factored part 33.55%

(3186700865+4*61959394)*653#+1 is prime! (0.0232s+0.0013s)

Primality testing (3186700865+5*61959394)*653#+1 [N-1, Brillhart-Lehmer-Selfridge]

Running N-1 test using base 23

Calling Brillhart-Lehmer-Selfridge with factored part 33.44%

(3186700865+5*61959394)*653#+1 is prime! (0.0228s+0.0013s)

Primality testing (3186700865+6*61959394)*653#+1 [N-1, Brillhart-Lehmer-Selfridge]

Running N-1 test using base 31

Calling Brillhart-Lehmer-Selfridge with factored part 33.55%

(3186700865+6*61959394)*653#+1 is prime! (0.0239s+0.0013s)

Primality testing (3186700865+7*61959394)*653#+1 [N-1, Brillhart-Lehmer-Selfridge]

Running N-1 test using base 43

Calling Brillhart-Lehmer-Selfridge with factored part 33.65%

(3186700865+7*61959394)*653#+1 is prime! (0.0228s+0.0014s)

Primality testing (3186700865+8*61959394)*653#+1 [N-1, Brillhart-Lehmer-Selfridge]

Running N-1 test using base 47

Calling Brillhart-Lehmer-Selfridge with factored part 34.29%

(3186700865+8*61959394)*653#+1 is prime! (0.0234s+0.0015s)

Primality testing (3186700865+9*61959394)*653#+1 [N-1, Brillhart-Lehmer-Selfridge]

Running N-1 test using base 59

Calling Brillhart-Lehmer-Selfridge with factored part 33.55%

(3186700865+9*61959394)*653#+1 is prime! (0.0249s+0.0014s)

Primality testing (3186700865+10*61959394)*653#+1 [N-1, Brillhart-Lehmer-Selfridge]

Running N-1 test using base 67

Calling Brillhart-Lehmer-Selfridge with factored part 33.44%

(3186700865+10*61959394)*653#+1 is prime! (0.0237s+0.0020s) - Ken wrote:
> when extended downwards turned out to be not only an AP10 but

Congratulations!

> also an AP11 giving me two records at once

>

> The following descibes a AP11 with 282-283 digits

>

> (3186700865+n*61959394)*653#+1 n= 0-10

That lucky AP11 looks hard to beat.

http://users.cybercity.dk/~dsl522332/math/aprecords.htm is updated.

--

Jens Kruse Andersen - --- In primeform@yahoogroups.com,

"kraDen" <kradenken@...> wrote:

> not only an AP10 but also an AP11

Oh, that is very neat, Ken.

> giving me two records at once

Poisson is both a friend and a foe. Sometimes we wait

much longer than we might have hoped; sometimes far less

than we might have dared to hope.

I am particularly happy for you, in the

case of this AP11, since I know how keenly you

appreciate the ups and downs of fortune that

Poisson dictates, in the discovery of primality.

Best regards

David - --- In primeform@yahoogroups.com, "djbroadhurst" <d.broadhurst@...> wrote:
>

It might be nice to try and quantify how good this record is.

> --- In primeform@yahoogroups.com,

> "kraDen" <kradenken@> wrote:

>

> > not only an AP10 but also an AP11

> > giving me two records at once

>

> Oh, that is very neat, Ken.

>

> Poisson is both a friend and a foe. Sometimes we wait

> much longer than we might have hoped; sometimes far less

> than we might have dared to hope.

>

> I am particularly happy for you, in the

> case of this AP11, since I know how keenly you

> appreciate the ups and downs of fortune that

> Poisson dictates, in the discovery of primality.

Updating the table in my post "How hard is it to find an AP-k" at

http://tech.groups.yahoo.com/group/primenumbers/message/20368

gives this:-

k d log(d) s=(k+4)*log(d)

- - ------ --------------

3 159382 11.979 83.853

4 11961 9.3894 75.115

5 7009 8.8550 79.695

6 2153 7.6746 76.746

7 1335 7.1967 79.164

8 1057 6.9632 83.558

9 425 6.0521 78.677

10 283 5.6454 79.036

11 283 5.6454 84.682

12 173 5.1533 82.453

13 78 4.3567 74.064

14 71 4.2627 76.728

15 54 3.9890 75.791

16 42 3.7377 74.753

17 42 3.7377 78.491

18 29 3.3673 74.081

19 27 3.2958 75.803

20 21 3.0445 73.068

21 20 2.9957 74.893

22 19 2.9444 76.554

23 19 2.9444 79.499

24 18 2.8904 80.930

25 18 2.8904 83.821

26 18 2.8904 86.711

and assigns second place to this amazing AP11, behind the recently-discovered first-ever AP26 (which took thousands of CPU days to find).

David's AP3 of a few days ago is given 3rd place.

It seems rather satisfactory that this heuristic formula allocates credit in a not unreasonable way over the entire spectrum of k values.

At the very least it is a "smoother" of the current record values, and would indicate in particular that Jaroslav's AP20, Jens's AP18 and my AP13 may be ripe for improvement...

Mike