- For this 4135-digit triplet record, Pfgw prompted

for the extra bits -x5231887, which was quicker

than bridging such a small BLS deficit by ECM effort,

or by invoking Konyagin-Pomerance.

Here is the log:

<<

Primality testing

(39553075974*4436*3251#*(4436*3251#+1)+210)*(4436*3251#-1)/35 + 5

[N-1/N+1, Brillhart-Lehmer-Selfridge]

Calling N+1 BLS with factored part 33.50%

and helper 0.01% (100.51% proof)

(39553075974*4436*3251#*(4436*3251#+1)+210)*(4436*3251#-1)/35 + 5

is prime! (82.639000 seconds)

Primality testing

(39553075974*4436*3251#*(4436*3251#+1)+210)*(4436*3251#-1)/35 + 7

[N-1/N+1, Brillhart-Lehmer-Selfridge]

Calling N-1 BLS with factored part 33.50%

and helper 0.01% (100.51% proof)

(39553075974*4436*3251#*(4436*3251#+1)+210)*(4436*3251#-1)/35 + 7

is prime! (35.982000 seconds)

Primality testing

(39553075974*4436*3251#*(4436*3251#+1)+210)*(4436*3251#-1)/35 + 11

[N-1/N+1, Brillhart-Lehmer-Selfridge]

Calling N+1 BLS with factored part 33.28%

and helper 0.01% (99.85% proof)

Proof incomplete rerun with -x5231887

(39553075974*4436*3251#*(4436*3251#+1)+210)*(4436*3251#-1)/35 + 11

is Fermat and Lucas PRP! (18.246000 seconds)

[doing as suggested:]

Primality testing

(39553075974*4436*3251#*(4436*3251#+1)+210)*(4436*3251#-1)/35 + 11

[N-1/N+1, Brillhart-Lehmer-Selfridge]

Calling N+1 BLS with factored part 33.28%

and helper 0.01% (99.85% proof)

1/5231887

8193/5231887

[snip 600 similar progress reports]

5210113/5231887

5218305/5231887

5226497/5231887

(39553075974*4436*3251#*(4436*3251#+1)+210)*(4436*3251#-1)/35 + 11

is prime! (579.723000 seconds)

[and finally here are the proofs of the small

twins that seed this record triplet:]

Primality testing 4436*3251#+1

[N-1/N+1, Brillhart-Lehmer-Selfridge]

Calling N-1 BLS with factored part 100.00%

and helper 0.57% (300.59% proof)

4436*3251#+1 is prime! (6.700000 seconds)

Primality testing 4436*3251#-1

[N-1/N+1, Brillhart-Lehmer-Selfridge]

Calling N+1 BLS with factored part 100.00%

and helper 0.37% (300.42% proof)

4436*3251#-1 is prime! (18.717000 seconds)

>>

David Broadhurst

- Hmm. That record lasted only for hours;

this 5-7-11 triplet is a mite bigger:

Primality testing

(90159302514*4436*3251#*(4436*3251#+1)+210)*(4436*3251#-1)/35 + 5

[N-1/N+1, Brillhart-Lehmer-Selfridge]

Calling N+1 BLS with factored part 33.33%

and helper 0.02% (100.02% proof)

(90159302514*4436*3251#*(4436*3251#+1)+210)*(4436*3251#-1)/35 + 5

is prime! (71.673000 seconds)

Primality testing

(90159302514*4436*3251#*(4436*3251#+1)+210)*(4436*3251#-1)/35 + 7

[N-1/N+1, Brillhart-Lehmer-Selfridge]

Calling N-1 BLS with factored part 33.33%

and helper 0.39% (100.39% proof)

(90159302514*4436*3251#*(4436*3251#+1)+210)*(4436*3251#-1)/35 + 7

is prime! (37.424000 seconds)

Primality testing

(90159302514*4436*3251#*(4436*3251#+1)+210)*(4436*3251#-1)/35 + 11

[N-1/N+1, Brillhart-Lehmer-Selfridge]

Calling N+1 BLS with factored part 33.49%

and helper 0.03% (100.50% proof)

(90159302514*4436*3251#*(4436*3251#+1)+210)*(4436*3251#-1)/35 + 11

is prime! (18.306000 seconds)

with no extra bits needed.

I still wait for a 7-11-13 triplet, at 4000+ digits,

to exemplify the cubic Ansatz more fully.

David Broadhurst - I had lively hopes when noting that

> I still wait for a 7-11-13 triplet, at 4000+ digits,

Not long to wait, in fact:

> to exemplify the cubic Ansatz more fully.

<<

Primality testing

(18599651274*4436*3251#*(4436*3251#+1)+210)*(4436*3251#-1)/35 + 7

[N-1/N+1, Brillhart-Lehmer-Selfridge]

Calling N-1 BLS with factored part 33.34%

and helper 0.25% (100.29% proof)

(18599651274*4436*3251#*(4436*3251#+1)+210)*(4436*3251#-1)/35 + 7

is prime! (33.098000 seconds)

Primality testing

(18599651274*4436*3251#*(4436*3251#+1)+210)*(4436*3251#-1)/35 + 11

[N-1/N+1, Brillhart-Lehmer-Selfridge]

Calling N+1 BLS with factored part 33.28%

and helper 0.15% (100.00% proof)

1/4

(18599651274*4436*3251#*(4436*3251#+1)+210)*(4436*3251#-1)/35 + 11

is prime! (18.286000 seconds)

Primality testing

(18599651274*4436*3251#*(4436*3251#+1)+210)*(4436*3251#-1)/35 + 13

[N-1/N+1, Brillhart-Lehmer-Selfridge]

Calling N-1 BLS with factored part 33.28%

and helper 0.04% (99.89% proof)

Proof incomplete rerun with -x123013

(18599651274*4436*3251#*(4436*3251#+1)+210)*(4436*3251#-1)/35 + 13

is Fermat and Lucas PRP! (22.422000 seconds)

[rerun as instructed:]

Primality testing

(18599651274*4436*3251#*(4436*3251#+1)+210)*(4436*3251#-1)/35 + 13

[N-1/N+1, Brillhart-Lehmer-Selfridge]

Calling N-1 BLS with factored part 33.28%

and helper 0.04% (99.89% proof)

1/2460259

8193/2460259

16385/2460259

24577/2460259

32769/2460259

40961/2460259

49153/2460259

57345/2460259

65537/2460259

73729/2460259

81921/2460259

90113/2460259

98305/2460259

106497/2460259

114689/2460259

122881/2460259

(18599651274*4436*3251#*(4436*3251#+1)+210)*(4436*3251#-1)/35 + 13

is prime! (36.102000 seconds)

>>

But it's smaller than yesterday's 5-7-11 records...

David Broadhurst - And, for good measure, another 7-11-13 BLS Triplet

which is now the current Triplet record holder:

<<

Primality testing

(108748629354*4436*3251#*(4436*3251#+1)+210)*(4436*3251#-1)/35 + 7

[N-1/N+1, Brillhart-Lehmer-Selfridge]

Calling N-1 BLS with factored part 33.36%

and helper 0.15% (100.26% proof)

(108748629354*4436*3251#*(4436*3251#+1)+210)*(4436*3251#-1)/35 + 7

is prime! (32.857000 seconds)

Primality testing

(108748629354*4436*3251#*(4436*3251#+1)+210)*(4436*3251#-1)/35 + 11

[N-1/N+1, Brillhart-Lehmer-Selfridge]

Calling N+1 BLS with factored part 33.28%

and helper 0.14% (99.96% proof)

1/94

(108748629354*4436*3251#*(4436*3251#+1)+210)*(4436*3251#-1)/35 + 11

is prime! (21.782000 seconds)

Primality testing

(108748629354*4436*3251#*(4436*3251#+1)+210)*(4436*3251#-1)/35 + 13

[N-1/N+1, Brillhart-Lehmer-Selfridge]

Calling N-1 BLS with factored part 33.28%

and helper 0.28% (100.11% proof)

(108748629354*4436*3251#*(4436*3251#+1)+210)*(4436*3251#-1)/35 + 13

is prime! (21.060000 seconds)

>>

Now I should trawl my 1-5-7-11-13 quintuply-sieved

database, already PFGWed at 7, in case there is a

record-breaking 1-7-13 CPAP3 lurking there too.

[Current CPAP3 record is merely 3161 digits.]

Note that a CPAP3 has a probability only 1/3 of

of that for finding a triplet. And I was lucky

to get 4 triplets. No harm in hoping...

David Broadhurst - --- In primeform@y..., "David Broadhurst" <d.broadhurst@o...> wrote:
> And, for good measure, another 7-11-13 BLS Triplet

David , daily -- even hourly -- you are re-writing the top triplet

> which is now the current Triplet record holder:

>

records. Will there be a fifth by sunset?

...

> And I was lucky

Nice work! Add diligence to your luck!

> to get 4 triplets. No harm in hoping...

Paul