## 5132 digit BLS provable AP5

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• Hi All, My search for 5000 digit BLS provable CPAP#/triplets is nearing its end. Over the past 4.5 years I have used approx 20GHZ years processing primes of
Message 1 of 2 , Feb 9, 2007
Hi All,
My search for >5000 digit BLS provable CPAP#/triplets is nearing its
end.
Over the past 4.5 years I have used approx 20GHZ years processing
primes of the type
(N*205881*4001#*(205881*4001#+1)+210)*(205881*4001#-1)/35+7
I have prped over 90,000,000 numbers.
Found 276228 prps which contained 2562 pairs (+7 and one of +1 +5 +11
and +13)
(8 triples (2CPAP3's,4 triplets and 2 of no use)
As a sub search I started looking for arithmetic progressions.
I have found
2123751 AP3s
2173 Ap4s
and 2 ( the second one being the reason for this post) AP5's.
(49077426729*205881*4001#*(205881*4001#+1)+210)*(205881*4001#-1)/35+7
+ N*(681402540*205881*4001#*(205881*4001#+1)*(205881*4001#-1)/35)
is prime for N=(0-4)

All proofs used -tp
additional -xvalues were required for N=1 and N=3

Primality testing (49077426729*205881*4001#*(205881*4001#+1)+210)*
(205881*4001#-1)/35+7 +

0*(681402540*205881*4001#*(205881*4001#+1)*(205881*4001#-1)/35) [N-1,
Brillhart-Lehmer-Selfridge]
Running N-1 test using base 3
Running N-1 test using base 5
Calling Brillhart-Lehmer-Selfridge with factored part 33.35%
(49077426729*205881*4001#*(205881*4001#+1)+210)*(205881*4001#-1)/35+7
+ 0*(681402540*205881*4001#*(205881*4001#+1)*(205881*4001#-1)/35) is

prime! (27.2622s+0.1807s)

Using -x8177729

Primality testing (49077426729*205881*4001#*(205881*4001#+1)+210)*
(205881*4001#-1)/35+7 +

1*(681402540*205881*4001#*(205881*4001#+1)*(205881*4001#-1)/35) [N-
1/N+1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 3
Running N-1 test using base 5
Running N-1 test using base 11
Running N+1 test using discriminant 19, base 1+sqrt(19)
Calling N-1 BLS with factored part 33.25% and helper 0.11% (99.87%
proof)
1/571185908
8193/571185908
16385/571185908
lots of lines removed
8167425/571185908
8175617/571185908
(49077426729*205881*4001#*(205881*4001#+1)+210)*(205881*4001#-1)/35+7
+ 1*(681402540*205881*4001#*(205881*4001#+1)*(205881*4001#-1)/35) is

prime! (1722.9017s+0.0469s)

Primality testing (49077426729*205881*4001#*(205881*4001#+1)+210)*
(205881*4001#-1)/35+7 +

2*(681402540*205881*4001#*(205881*4001#+1)*(205881*4001#-1)/35) [N-
1/N+1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 23
Running N-1 test using base 31
Running N+1 test using discriminant 43, base 3+sqrt(43)
Calling N-1 BLS with factored part 33.37% and helper 0.02% (100.13%
proof)
(49077426729*205881*4001#*(205881*4001#+1)+210)*(205881*4001#-1)/35+7
+ 2*(681402540*205881*4001#*(205881*4001#+1)*(205881*4001#-1)/35) is

prime! (40.5022s+0.2313s)

using -x76085

Primality testing (49077426729*205881*4001#*(205881*4001#+1)+210)*
(205881*4001#-1)/35+7 +

3*(681402540*205881*4001#*(205881*4001#+1)*(205881*4001#-1)/35) [N-
1/N+1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 3
Running N+1 test using discriminant 11, base 2+sqrt(11)
Calling N-1 BLS with factored part 33.27% and helper 0.09% (99.91%
proof)
1/1034114599
8193/1034114599
16385/1034114599
a few lines removed
65537/1034114599
73729/1034114599
(49077426729*205881*4001#*(205881*4001#+1)+210)*(205881*4001#-1)/35+7
+ 3*(681402540*205881*4001#*(205881*4001#+1)*(205881*4001#-1)/35) is

prime! (53.8161s+0.0648s)

Primality testing (49077426729*205881*4001#*(205881*4001#+1)+210)*
(205881*4001#-1)/35+7 +

4*(681402540*205881*4001#*(205881*4001#+1)*(205881*4001#-1)/35) [N-
1/N+1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 59
Running N-1 test using base 67
Running N+1 test using discriminant 79, base 4+sqrt(79)
Calling N-1 BLS with factored part 33.34% and helper 0.11% (100.15%
proof)
(49077426729*205881*4001#*(205881*4001#+1)+210)*(205881*4001#-1)/35+7
+ 4*(681402540*205881*4001#*(205881*4001#+1)*(205881*4001#-1)/35) is

prime! (39.7342s+0.3566s)

I plan to finish up my search by extending the 2123751 AP3's to see
if I can find any more Ap5s then I'll start a new project.
Cheers
Ken
• ... Congratulations on the results of this massive effort. The new AP5 record is at http://hjem.get2net.dk/jka/math/aprecords.htm which also contains many
Message 2 of 2 , Feb 9, 2007
Ken Davis wrote:
> My search for >5000 digit BLS provable CPAP#/triplets is nearing
> its end.
> Over the past 4.5 years I have used approx 20GHZ years

> (49077426729*205881*4001#*(205881*4001#+1)+210)*(205881*4001#-1)/35+7
> + N*(681402540*205881*4001#*(205881*4001#+1)*(205881*4001#-1)/35)
> is prime for N=(0-4)

Congratulations on the results of this massive effort.
The new AP5 record is at http://hjem.get2net.dk/jka/math/aprecords.htm
which also contains many recent long AP records by Jaroslaw Wroblewski.

Thanks for including APTreeSieve in the Prime Pages prover code.
Sorry I haven't made a publicly available version.
It may still happen at some time if there is interest.

--
Jens Kruse Andersen
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