Hi Joss and Masser,

thank's for your interest. First of all I should note that I'm currently out of town for about one week. Therefore I haven't full access to all the data I'd like to provide to you.

For the Nash weight computation I used a small C program written by myself after a "close" look into John Brennen's Java applet. I developed my program in a UNIX environment using the GMP library. I compiled and used it under Linux too, but I don't know if GMP is available for Windows and how to get it compiled there. Nevertheless, if some of you are interested, I could send you the source code and/or Linux binary.

Please don't ask for a "complete" list of Nash weights for all (odd) k's up to 350 million. Since my program stores the results into ascii files this would be 3.5 GByte of data (10.5 MByte per 1 million range of k's = 500.000 odd k's)! But any small portion of data can be easily extracted for the list, e.g. the "less than 100" list or a "larger than 6000" list or a "zero weight" list (= Riesel numbers). The "less than 100" list would be about 900 kByte (or 300 kByte ZIPed).

Originally I had planned to take one or a few k's up to 10M and I did a lot of sieving work in that direction. But after all it is too much work for one single person having only a few machines. So any of you may take part in the search for some big primes ...

The following 23 k values have been sieved for n=2..10.000.000 up to p=8.000.000.000.000 (yes, 8 trillion!) and already LLR-tested up to n=1.000.000:

Code:

k w w' prime for n=
-----------------------------------------
19370947 17 15 25
59910449 15 15 92
80857169 19 17 4
143316643 15 16
162405629 20 21 896, 12236
175437131 20 21
189030223 18 19
203012861 16 17 754
209826493 16 18
224371169 17 17 3548
243163663 15 18 919087
245265883 16 17
260213857 13 16
265831619 19 16
276278983 16 18 21623, 473423
290851087 23 21 57
298095191 14 15
300871183 22 23
308120317 19 19
308141737 19 19 7517
315940139 18 17 8388, 595620
326840893 17 15 31
----------------------------------------
(w is the "original" Nash weight for n=100.001-110.000,
while w' is the Nash weight obtained for n=1-10.000)

I have no idea, which k should be choosen for testing n>1 million. Maybe one or a few for which no prime exists for n<1 million.

As I already mentioned, I don't have access to the presieved ranges at the moment, but I can send them to Joss around March 15th, so he could store the data on 15k.org.

It should be mentioned that a single LLR test on a n=1 million number takes about 4 hours on a 2.4 GHz P4. And around n=1.5 million the time raises to 8 hours and more.

In case you can't wait until I've sent you the presieved data and/or you want to test some untested k-values for lower n, here is a short list of low weight k's which I haven't tested so far (but someone else may have tested them):

Code:

k w w'
----------------------
10013593 40 46
10247561 38 38
10284899 27 30
10346561 46 48
10453199 28 26
10463923 34 28
10598947 45 45
10639619 37 38
10671431 34 36
10805593 49 45
10813783 34 40
10906603 38 38
10932097 44 41
10943321 50 61
11223059 26 27
11311003 31 28
11319193 45 43
11468609 46 47
11639819 39 38
11658187 48 44
11716993 48 41
11741347 42 44
11846279 50 48
11847299 45 45
11932211 38 37
11955659 36 36
12254533 36 36
12305981 44 44
12334093 49 53
12551291 43 41
12695159 31 36
12753311 48 42
12825793 49 53
13277471 48 50
13651423 48 49
13768087 46 49
13900807 23 18
14085941 35 36
14321533 29 23
14466883 40 41
14533213 42 44
14549347 48 52
14745343 31 32
14961487 25 23
----------------------

I suggest to sieve them for n=2..250.000 up to about p=10 or 15 billion. Only 150-400 candidates will remain per k and can be LLR tested in a few hours. Candidates for n<256 should be tested by Pari (or Mathematica, Maple, etc.) rather than using LLR, since it may crash or report a prime as "non-prime".

There should be quite a few primes in the above list for n<250.000 but only

one or two Top5000 primes.

Best wishes and a few primes ...

Thomas