I have completed the (near-)Woodall [I]k[/I]'s, except for [I]k[/I]=1993191, to [I]n[/I]=350k. I found 2 primes:[list][*]1268979*2^332833-1[*]197673*2^341268-1[/list]
pepi37 kindly ran a large chunk of the range, and he found the following 2 primes:[list][*]251749*2^333881-1[*]1467763*2^340379-1[/list]
The list of [I]k[/I]'s tested can be found in the previous post.

k = 50171
 

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

k = 1549573
 

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

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.

[QUOTE=Viliam Furik;554266]
So far I have checked up to n = 395 000 and found 4 primes
[/QUOTE]

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

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. ;)

1281979*2^n-1 has a [url='https://www.rieselprime.de/ziki/Nash_weight']Nash weight[/url] of 1789, so relatively low.
1281979 [tex]\equiv[/tex] 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 [url='https://www.rieselprime.de/ziki/Liskovets-Gallot_conjectures']Liskovets-Gallot conjectures[/url] 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.

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.

[QUOTE]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[/QUOTE]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.

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 [url='https://www.rieselprime.de/ziki/Riesel_k%3DLow_weight']low weight[/url] 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.

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:

[CODE]1281979 * 2^n - 1
0 <= n <= 20000

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


bold values indicate primes[/CODE]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.
[QUOTE]You could choose a lower k-value which produce smaller test timings for same n-values as 1281979.[/QUOTE]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 ... ;)

[QUOTE=bur;556433]So the number of primes per n decreases stronger than the number of candidates per n?)[/QUOTE]

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