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## New Ap10 and Ap11 records

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• 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
Message 1 of 4 , Apr 29, 2010
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)
• ... Congratulations! That lucky AP11 looks hard to beat. http://users.cybercity.dk/~dsl522332/math/aprecords.htm is updated. -- Jens Kruse Andersen
Message 2 of 4 , Apr 30, 2010
Ken wrote:
> 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

Congratulations!
That lucky AP11 looks hard to beat.
http://users.cybercity.dk/~dsl522332/math/aprecords.htm is updated.

--
Jens Kruse Andersen
• ... 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
Message 3 of 4 , Apr 30, 2010
--- 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.

Best regards

David
• ... It might be nice to try and quantify how good this record is. Updating the table in my post How hard is it to find an AP-k at
Message 4 of 4 , May 1, 2010
--- In primeform@yahoogroups.com, "djbroadhurst" <d.broadhurst@...> wrote:
>
> --- 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.

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

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
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