Choose your own K and work on finding a top-5000 prime!

[Happy5214]

I have completed the (near-)Woodall k's, except for k=1993191, to n=350k. I found 2 primes:

• 1268979*2^332833-1
• 197673*2^341268-1

pepi37 kindly ran a large chunk of the range, and he found the following 2 primes:

• 251749*2^333881-1
• 1467763*2^340379-1

The list of k's tested can be found in the previous post.

[Dylan14]

k = 50171 is at n = 2.945 M, no primes found, continuing...

[Happy5214]

k=1549573 has now been tested to n=300k. The known prime at n=260199 (on Prime Pages) was re-confirmed, and no new primes were found. I am releasing this k. The LLR log is attached (it's version 3.8.24, so PRP residues instead of LLR).

[Viliam Furik]

I restarted the search for primes with k = 105105.

So far I have checked up to n = 395 000 and found 4 primes:
105105*2^328791-1
105105*2^333964-1
105105*2^358832-1
105105*2^387091-1

Also regarding k=20020913, I have found another prime:
20020913*2^862692-1

I will update those k's in Prime Wiki.

[Viliam Furik]

Originally Posted by Viliam Furik

So far I have checked up to n = 395 000 and found 4 primes

Silly me... 5 primes. I forgot to mention n = 260723.

[bur]

I was testing k = 1281979 for n<= 100000 and interestingly the first 10 n for which this is prime are prime themselves. After that unfortunately there's n = 1005 which obviously isn't. But the ratio of n being prime vs being composite continues to be very high for the remainder of the range.

Is that a known property of some k and is it known why it happens? Is there a connection to Mersenne?


Nothing like that on the Proth side, but there the n that yield primes seem to come in pairs of two that are close to each other (5-10% difference). Though it might just be the brain's inbuild pattern recognition going overboard. So I'll keep testing. If there's something the next n should be in the 3e5 range. ;)

[kar_bon]

1281979*2^n-1 has a Nash weight of 1789, so relatively low.
1281979 1 mod 3 so all primes of that sequence are only odd n-values.
k-values with 2 mod 3 can only produce primes with even n's, and k's divisible by 3 can produce primes with odd/even n-values.

1281979*2^n+1 has a low Nash weight of 847, so there should less primes for this sequence.

The Liskovets-Gallot conjectures study the contribution of odd/even n-values of such seqs.
There exits k-values which never produce primes for any n-value like the Riesel problem.

PS: If your're done you can list the prime n-values in this thread and I can include those in the Wiki, both sides (Proth /Riesel) possible. Don't forget to give the search limits then.

[bur]

kar_bon, I never really understood the Nash weight. It is an indicator for how many candidates remain after sieving? So I would think a low weight is good, since few candidates after sieving is favorable? What do I miss?

I don't know what is average number of primes in n < 1e5 range, but I think there were maybe 15 for this k. Is that so little? Proth was giving similar number of primes.

If your're done you can list the prime n-values in this thread and I can include those in the Wiki, both sides (Proth /Riesel) possible. Don't forget to give the search limits then

I'll do that. 0 <= n <= 1e5 is done for both Riesel/Proth. I don't have the results here, but will post this evening.


The main task I want to accomplish is finding a mega prime using Proth20, so I'm currently sieving 3320000 <= n <= 4100000 for Proth side. But I also plan to do the remaining smaller n values for both Riesel/Proth using LLR on CPU. I'll post them here in the future.

[kar_bon]

The less the Nash weight is the less candidates remain after sieving is correct, but also the less chance to find a prime: low Nash = less cand. = less primes.

You could choose a lower k-value which produce smaller test timings for same n-values as 1281979.
Check the Wiki for low weight k-values to see the difference. You could sieve and test some higher ranges to get a feeling of those.
Looking other tables in the Wiki and sorting by #primes or Nash could help, too.

[bur]

So the number of primes per n decreases stronger than the number of candidates per n?


Here are the n values that produce Riesel primes:

1281979 * 2^n - 1
0 <= n <= 20000

3
7
43
79
107
157
269
307
373
397
1005
1013
1765
1987
2269
6623
7083
7365
10199
16219


bold values indicate primes

I have completed the same for Proth side, but since it's a work in progress, I'll post once I'm at larger n.

You could choose a lower k-value which produce smaller test timings for same n-values as 1281979.

There's a longer story behind why I chose that k... :D

I have two computers at work I can use for crunching, both are using CPU for Primegrid projects. I want to keep it that way, because I like the conjecture solving. So the cheapest way to do more crunching would be to buy to mid-low-end GPUs such as the GTX 1650 and use those for other projects. Then I found out LLR on GPUs is considered a waste of time.

But - there's a new software Proth2.0 that apparently tests Proth primes quite efficiently on GPUs. So I decided to find Proth primes. But PG has three Proth prime subprojects and covers a lot of small k's... Around that time I discovered my birth date is a prime number and also large enough not to interfere with PG. Also that it's prime could result in the interesting combination of prime k, prime n and prime b. I know large k hardly change anything in regard to total digits but make computation slower and I also knew the Nash weight is not that high using nash.exe, but I will keep that k. If I find a mega prime with it it will at least be a somewhat rare k ... ;)

[VBCurtis]

Originally Posted by bur

So the number of primes per n decreases stronger than the number of candidates per n?)

Nope. But the opposite isn't true, either.