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## AP13s

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• Hi All, The following descibe AP13s with 83 digits (9773795412+\$n*81373857)*181#+1 n=0-12 (7967748420+\$n*340308346)*181#+1) n=0-12 Input/output statistics:-
Message 1 of 9 , May 19, 2010
Hi All,

The following descibe AP13s with 83 digits

(9773795412+\$n*81373857)*181#+1 n=0-12
(7967748420+\$n*340308346)*181#+1) n=0-12

Input/output statistics:-
Numbers input to NewPGen: 3*10^9 seived to 1*10^11
Nubers input to PFGW 626824792. PRP's found by PFGW: 148388269
AP13 search run against the PRPs in the first 4*10^6 of the numbers (exhaustively finding all AP in the 3*10^9 range)

I know some, Mike at least, appreciate stats so
AP07s:14260855
AP08s:601980
AP09s:26092
AP10s:1215
AP11s:58
AP12s:2
AP13s:0

Additional APs through extension of ap7'3 through 12's (both forward and back)gave additional
AP08s:773320
AP09s:69921
AP10s:4908
AP11s:288
AP12s:12
AP13s:2

First Ap13
Primality testing (9773795412+0*81373857)*181#+1 [N-1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 191
Calling Brillhart-Lehmer-Selfridge with factored part 35.77%
(9773795412+0*81373857)*181#+1 is prime! (0.0047s+0.0016s)
Primality testing (9773795412+1*81373857)*181#+1 [N-1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 2
Calling Brillhart-Lehmer-Selfridge with factored part 33.58%
(9773795412+1*81373857)*181#+1 is prime! (0.0042s+0.0017s)
Primality testing (9773795412+2*81373857)*181#+1 [N-1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 197
Running N-1 test using base 199
Calling Brillhart-Lehmer-Selfridge with factored part 33.94%
(9773795412+2*81373857)*181#+1 is prime! (0.0061s+0.0017s)
Primality testing (9773795412+3*81373857)*181#+1 [N-1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 3
Calling Brillhart-Lehmer-Selfridge with factored part 34.67%
(9773795412+3*81373857)*181#+1 is prime! (0.0043s+0.0017s)
Primality testing (9773795412+4*81373857)*181#+1 [N-1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 211
Running N-1 test using base 227
Calling Brillhart-Lehmer-Selfridge with factored part 33.58%
(9773795412+4*81373857)*181#+1 is prime! (0.0062s+0.0018s)
Primality testing (9773795412+5*81373857)*181#+1 [N-1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 7
Calling Brillhart-Lehmer-Selfridge with factored part 34.31%
(9773795412+5*81373857)*181#+1 is prime! (0.0047s+0.0019s)
Primality testing (9773795412+6*81373857)*181#+1 [N-1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 223
Calling Brillhart-Lehmer-Selfridge with factored part 33.58%
(9773795412+6*81373857)*181#+1 is prime! (0.0047s+0.0017s)
Primality testing (9773795412+7*81373857)*181#+1 [N-1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 11
Calling Brillhart-Lehmer-Selfridge with factored part 35.40%
(9773795412+7*81373857)*181#+1 is prime! (0.0046s+0.0017s)
Primality testing (9773795412+8*81373857)*181#+1 [N-1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 193
Running N-1 test using base 239
Calling Brillhart-Lehmer-Selfridge with factored part 35.40%
(9773795412+8*81373857)*181#+1 is prime! (0.0059s+0.0019s)
Primality testing (9773795412+9*81373857)*181#+1 [N-1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 19
Calling Brillhart-Lehmer-Selfridge with factored part 35.77%
(9773795412+9*81373857)*181#+1 is prime! (0.0044s+0.0017s)
Primality testing (9773795412+10*81373857)*181#+1 [N-1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 229
Calling Brillhart-Lehmer-Selfridge with factored part 35.04%
(9773795412+10*81373857)*181#+1 is prime! (0.0047s+0.0017s)
Primality testing (9773795412+11*81373857)*181#+1 [N-1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 23
Calling Brillhart-Lehmer-Selfridge with factored part 35.40%
(9773795412+11*81373857)*181#+1 is prime! (0.0044s+0.0017s)
Primality testing (9773795412+12*81373857)*181#+1 [N-1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 241
Calling Brillhart-Lehmer-Selfridge with factored part 34.67%
(9773795412+12*81373857)*181#+1 is prime! (0.0043s+0.0017s)

2nd Ap13
Primality testing (7967748420+0*340308346)*181#+1 [N-1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 191
Calling Brillhart-Lehmer-Selfridge with factored part 34.67%
(7967748420+0*340308346)*181#+1 is prime! (0.0044s+0.0016s)
Primality testing (7967748420+1*340308346)*181#+1 [N-1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 199
Calling Brillhart-Lehmer-Selfridge with factored part 35.40%
(7967748420+1*340308346)*181#+1 is prime! (0.0049s+0.0018s)
Primality testing (7967748420+2*340308346)*181#+1 [N-1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 197
Calling Brillhart-Lehmer-Selfridge with factored part 35.04%
(7967748420+2*340308346)*181#+1 is prime! (0.0048s+0.0015s)
Primality testing (7967748420+3*340308346)*181#+1 [N-1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 193
Calling Brillhart-Lehmer-Selfridge with factored part 34.31%
(7967748420+3*340308346)*181#+1 is prime! (0.0040s+0.0017s)
Primality testing (7967748420+4*340308346)*181#+1 [N-1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 229
Calling Brillhart-Lehmer-Selfridge with factored part 35.40%
(7967748420+4*340308346)*181#+1 is prime! (0.0046s+0.0018s)
Primality testing (7967748420+5*340308346)*181#+1 [N-1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 223
Calling Brillhart-Lehmer-Selfridge with factored part 35.40%
(7967748420+5*340308346)*181#+1 is prime! (0.0045s+0.0015s)
Primality testing (7967748420+6*340308346)*181#+1 [N-1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 233
Calling Brillhart-Lehmer-Selfridge with factored part 33.58%
(7967748420+6*340308346)*181#+1 is prime! (0.0045s+0.0014s)
Primality testing (7967748420+7*340308346)*181#+1 [N-1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 227
Calling Brillhart-Lehmer-Selfridge with factored part 34.31%
(7967748420+7*340308346)*181#+1 is prime! (0.0046s+0.0018s)
Primality testing (7967748420+8*340308346)*181#+1 [N-1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 239
Running N-1 test using base 241
Calling Brillhart-Lehmer-Selfridge with factored part 35.04%
(7967748420+8*340308346)*181#+1 is prime! (0.0058s+0.0015s)
Primality testing (7967748420+9*340308346)*181#+1 [N-1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 211
Calling Brillhart-Lehmer-Selfridge with factored part 33.94%
(7967748420+9*340308346)*181#+1 is prime! (0.0044s+0.0014s)
Primality testing (7967748420+10*340308346)*181#+1 [N-1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 197
Calling Brillhart-Lehmer-Selfridge with factored part 33.45%
(7967748420+10*340308346)*181#+1 is prime! (0.0044s+0.0014s)
Primality testing (7967748420+11*340308346)*181#+1 [N-1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 251
Calling Brillhart-Lehmer-Selfridge with factored part 33.82%
(7967748420+11*340308346)*181#+1 is prime! (0.0058s+0.0015s)
Primality testing (7967748420+12*340308346)*181#+1 [N-1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 233
Calling Brillhart-Lehmer-Selfridge with factored part 34.55%
(7967748420+12*340308346)*181#+1 is prime! (0.0047s+0.0015s)
Primality testing (7967748420+13*340308346)*181#+1 [N-1, Brillhart-Lehmer-Selfridge]

Cheers
Ken
• ... Many congrats, Ken! But I m puzzled about how this:- ... can give this:- ... Based on my http://tech.groups.yahoo.com/group/primeform/message/7672 I would
Message 2 of 9 , May 20, 2010
>
> The following descibe AP13s with 83 digits
>
> (9773795412+\$n*81373857)*181#+1 n=0-12
> (7967748420+\$n*340308346)*181#+1) n=0-12
>
> Input/output statistics:-
> Numbers input to NewPGen: 3*10^9 seived to 1*10^11
> Nubers input to PFGW 626824792. PRP's found by PFGW: 148388269
> AP13 search run against the PRPs in the first 4*10^6 of the numbers (exhaustively finding all AP in the 3*10^9 range)
>
> I know some, Mike at least, appreciate stats so
> AP07s:14260855
> AP08s:601980
> AP09s:26092
> AP10s:1215
> AP11s:58
> AP12s:2
> AP13s:0
>
> Additional APs through extension of ap7'3 through 12's (both forward and back)gave additional
> AP08s:773320
> AP09s:69921
> AP10s:4908
> AP11s:288
> AP12s:12
> AP13s:2

Many congrats, Ken!

But I'm puzzled about how this:-
> AP13 search run against the PRPs in the first 4*10^6 of the numbers

can give this:-
> AP12s:2

Based on my
http://tech.groups.yahoo.com/group/primeform/message/7672
I would have expected only about (4*10^6/1.05*10^8)^2*21= 0.03 AP12's from such a small range.

You didn't mean "the first 4*10^7" did you?

Mike
• Hi Mike, I did indeed mean 4*10^7 Sorry about that. Ken
Message 3 of 9 , May 20, 2010
Hi Mike,
I did indeed mean 4*10^7
Ken

--- In primeform@yahoogroups.com, "mikeoakes2" <mikeoakes2@...> wrote:
>
>
>
> --- In primeform@yahoogroups.com, "kraDen" <kradenken@> wrote:
> >
> > The following descibe AP13s with 83 digits
> >
> > (9773795412+\$n*81373857)*181#+1 n=0-12
> > (7967748420+\$n*340308346)*181#+1) n=0-12
> >
> > Input/output statistics:-
> > Numbers input to NewPGen: 3*10^9 seived to 1*10^11
> > Nubers input to PFGW 626824792. PRP's found by PFGW: 148388269
> > AP13 search run against the PRPs in the first 4*10^6 of the numbers (exhaustively finding all AP in the 3*10^9 range)
> >
> > I know some, Mike at least, appreciate stats so
> > AP07s:14260855
> > AP08s:601980
> > AP09s:26092
> > AP10s:1215
> > AP11s:58
> > AP12s:2
> > AP13s:0
> >
> > Additional APs through extension of ap7'3 through 12's (both forward and back)gave additional
> > AP08s:773320
> > AP09s:69921
> > AP10s:4908
> > AP11s:288
> > AP12s:12
> > AP13s:2
>
> Many congrats, Ken!
>
> But I'm puzzled about how this:-
> > AP13 search run against the PRPs in the first 4*10^6 of the numbers
>
> can give this:-
> > AP12s:2
>
> Based on my
> http://tech.groups.yahoo.com/group/primeform/message/7672
> I would have expected only about (4*10^6/1.05*10^8)^2*21= 0.03 AP12's from such a small range.
>
> You didn't mean "the first 4*10^7" did you?
>
> Mike
>
• ... Congratulations! http://users.cybercity.dk/~dsl522332/math/aprecords.htm is updated. -- Jens Kruse Andersen
Message 4 of 9 , May 20, 2010
Ken wrote:
> The following descibe AP13s with 83 digits
>
> (9773795412+\$n*81373857)*181#+1 n=0-12
> (7967748420+\$n*340308346)*181#+1) n=0-12

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

--
Jens Kruse Andersen
• ... Interesting question (about which I have no idea): what else might be done with a database 40 million 83-digit primes? Regarding my recent study of primes
Message 5 of 9 , May 20, 2010
--- In primeform@yahoogroups.com,

> I did indeed mean 4*10^7

Interesting question (about which I have no idea):
what else might be done with a database 40 million
83-digit primes?

Regarding my recent study of primes below 40 digits,
I am now able to report (after several haphazard
rectangular searches) that

23889376170*265^n+1 is prime for n=0..10

p(n)=23889376170*265^n+1;
for(n=0,10,if(isprime(p(n)),print(p(n))))

23889376171
6330684685051
1677631441538251
444572332007636251
117811667982023606251
31220092015236255656251
8273324384037607748906251
2192430961769966053460156251
580994204869041004166941406251
153963464290295866104239472656251
40800318036928404517623460253906251

It seems to me to be rather hard to find more
than 11 primes p[n] with p[n]-1 forming a
geometric progression with a ratio b > 265.

But I bet Jarek could do so, if he set his mind to it...

David
• ... I was wondering if there was any possibility of finding 11 (or 12) successive n values such that p(n) is prime for a very large (or large) n value. Is
Message 6 of 9 , May 23, 2010

> 23889376170*265^n+1 is prime for n=0..10
>
> p(n)=23889376170*265^n+1;
> for(n=0,10,if(isprime(p(n)),print(p(n))))
>
> 23889376171
> 6330684685051
> 1677631441538251
> 444572332007636251
> 117811667982023606251
> 31220092015236255656251
> 8273324384037607748906251
> 2192430961769966053460156251
> 580994204869041004166941406251
> 153963464290295866104239472656251
> 40800318036928404517623460253906251
>
> It seems to me to be rather hard to find more
> than 11 primes p[n] with p[n]-1 forming a
> geometric progression with a ratio b > 265.
>
> But I bet Jarek could do so, if he set his mind to it...
>
> David
>
I was wondering if there was any possibility of finding 11 (or 12) successive n values such that p(n) is prime for a very large (or large) n value. Is there more (or less?) probability to find long prime p(n) chains with n beginning at n=0 or with n>=n0 for arbitrary n0 values?
• In other words: 23889376170*265^n+1 is prime for n=0..10 Is there any (good enough) probability to find another 11 prime chain with n=n0..n0+10 for a given
Message 7 of 9 , May 23, 2010
In other words:
23889376170*265^n+1 is prime for n=0..10
Is there any (good enough) probability to find another 11 prime chain with n=n0..n0+10 for a given base?
Is there any probability that exists a 11 length (or more) prime chain k*b^n+c not discovered because n=n0..n0+10 with n0>0 ?
• ... The case n = n0 = 0 is subsumed by my case, with n = 0, Suppose that k*b^n + 1 is prime in the L cases n = n0 .. n0+L-1, then K*b^n + 1 is prime in the
Message 8 of 9 , May 23, 2010
--- In primeform@yahoogroups.com,

> Is there more (or less?) probability to find long prime p(n)
> chains with n beginning at n=0 or with n>=n0 for arbitrary n0.

The case n >= n0 >= 0 is subsumed by my case, with n >= 0,
Suppose that k*b^n + 1 is prime in the L cases n = n0 .. n0+L-1,
then K*b^n + 1 is prime in the L cases n = 0 .. L-1
where K = k*b^n0.

Reminder: The challenge is

> find more than 11 primes p[n] with p[n]-1 forming a
> geometric progression with a ratio b > 265

David
• ... OK, seen it.
Message 9 of 9 , May 23, 2010
>
>
>
> --- In primeform@yahoogroups.com,
> "ajo" <sopadeajo2001@> wrote:
>
> > Is there more (or less?) probability to find long prime p(n)
> > chains with n beginning at n=0 or with n>=n0 for arbitrary n0.
>
> The case n >= n0 >= 0 is subsumed by my case, with n >= 0,
> Suppose that k*b^n + 1 is prime in the L cases n = n0 .. n0+L-1,
> then K*b^n + 1 is prime in the L cases n = 0 .. L-1
> where K = k*b^n0.
>
> Reminder: The challenge is
>
> > find more than 11 primes p[n] with p[n]-1 forming a
> > geometric progression with a ratio b > 265
>
> David
>
OK, seen it.
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