Bases 501-1030 reservations/statuses/primes

[TimSorbet]

Originally Posted by Mini-Geek

PFGWing k=258 now...

Also no primes to n=10k. Unreserving Riesel 811.

[kar_bon]

tested just for fun the remaining k=2 from n=25000 and found this:

Primality testing 2*989^26868-1 [N+1, Brillhart-Lehmer-Selfridge]
Running N+1 test using discriminant 23, base 8+sqrt(23)
Calling Brillhart-Lehmer-Selfridge with factored part 54.54%
2*989^26868-1 is prime! (873.8538s+0.0080s)

so this is my first proven base (very small one but a beginning )

[gd_barnes]

Originally Posted by kar_bon

tested just for fun the remaining k=2 from n=25000 and found this:

Primality testing 2*989^26868-1 [N+1, Brillhart-Lehmer-Selfridge]
Running N+1 test using discriminant 23, base 8+sqrt(23)
Calling Brillhart-Lehmer-Selfridge with factored part 54.54%
2*989^26868-1 is prime! (873.8538s+0.0080s)

so this is my first proven base (very small one but a beginning )

Personally I would "give credit" for a base as "proven" if the final prime is n>10K so there you go, you got one! Well done. Below that and it just becomes "not particularly exciting" or noteworthy. Otherwise KEP and I would have tons of bases proven, most with a highest prime of n<100.

One thing of interest about this base: Although there are many bases with a conjecture of k=4 that are proven with only a prime for k=2, this is the 1st such base proven with a prime at n>250. The current bases proven by finding only a prime for k=2 at n>100 are:

Riesel base 989, n=26868
Riesel base 779, n=220
Riesel base 629, n=186
Riesel base 449, n=174
Riesel base 29, n=136


All bases where b==(29 mod 30) will have a conjecture of k=4 on both sides and will only need to be tested for k=2 because odd k's will have a trivial factor of 2. Afaik, all of the Riesel bases <= 1024 have been done but most of the higher Sierp bases have not been. If anyone wants to take on the task of doing some of them, most can be done almost instantly and will have a prime at n<10. If you decide to do this, please let me know ahead of time. Most will likely test very quickly but multiple bases take quite a while to add to the pages even for a small conjecture. I'll want to know which bases so I can start adding them to the pages before getting all of the info.

Edit: There is only one such Riesel base <= 1024 remaining to be proven. The highest one: base 1019, which has currently been tested to n=25K. So there you go Karsten...another possible one to prove. Doing so would prove all Riesel bases <= 1024 where b==(29 mod 30). :-)


Gary

[Batalov]

I'll take Sierp. base 961 to 50K, for starters.

[Batalov]

Sierp. base 961 around 15K; 3 primes, 9 to go.

[gd_barnes]

Originally Posted by Batalov

Sierp. base 961 around 15K; 3 primes, 9 to go.

I can't change the testing limit on the pages without a listing of the primes found. I'll just leave it at n=2500 for now.

[Batalov]

I'll take Riesel base 811.

[Batalov]

Riesel base 811 is proven:
8*811^31783-1 is 3-PRP! (189.6443s+0.0104s)
258*811^28010-1 is 3-PRP! (179.3691s+0.0093s)

Running N+1 test using discriminant 3, base 1+sqrt(3)
Special modular reduction using FFT length 32K on 8*811^31783-1
Calling Brillhart-Lehmer-Selfridge with factored part 100.00%
8*811^31783-1 is prime! (2419.9390s+0.0107s)

Running N+1 test using discriminant 3, base 3+sqrt(3)
Special modular reduction using zero-padded FFT length 48K on 258*811^28010-1
...running... will let you know if it's not prime.

[gd_barnes]

Great work Serge. Another one bites the dust!

Those b==(1 mod 30) bases sure are prime.

[Batalov]

I am running a lot of odd high bases, just to get a feeling.
I will carefully catalog what's there, and the ranges and result files, but for now, some cleared k's to include into the webpage:

508*31^7188+1 is 3-PRP! (1.4761s+0.0008s) <== that's Sierp base 961
586*31^15728+1 is 3-PRP! (6.6518s+0.0013s)
636*31^8674+1 is 3-PRP! (2.3250s+0.0008s)
120*601^4663-1 is 3-PRP! (4.1509s+0.0024s)
378*811^6792+1 is 3-PRP! (7.5817s+0.0011s)

[gd_barnes]

Originally Posted by Batalov

Riesel base 811 is proven:
8*811^31783-1 is 3-PRP! (189.6443s+0.0104s)
258*811^28010-1 is 3-PRP! (179.3691s+0.0093s)

An amazing note about this proof:

This is only the 3rd base proven with TWO primes of n>25K and the very 1st Riesel base! The other two are Sierp bases 11 and 23, the latter of which is the only one with two primes of n>100K.

What's so remarkable is that the base is 35X larger than any previous base with this attribute!

There is only one k remaining on 9 bases that would end up having 3 or more primes of n>25K if we can get them proven. They are Riesel bases 22, 23, 27, 49, and 72 and Sierp bases 9, 10, 17, and 33.

If proven, Riesel base 22 would have 5 primes of n>25K and Sierp base 17 would have 4. All the rest above would have 3. Sierp base 17 would be the 1st one with 3 primes of n>100K!


Gary