20040306, 01:27  #1 
Sep 2002
106_{16} Posts 
Non 15k to 1M anyone?
As you all know 15k's usually have more primes in the lower n's. More primes also mean more n's to test. I have been working for past couple weeks to find k's that are easier to sieve. They have very few primes but also have very few n's to test thus faster the way to 1M. The best one that came up so far is 79*487 (38473) it has only 2049 n's left to be tested from 300k to 1M. The funny thing is, I haven't found one yet! Would any be interested in joining a team to take a non 15k to 1M? It could be after the current 15k team!
Joss L15 
20040306, 15:08  #2 
Feb 2003
774_{16} Posts 
I'm currently testing a few non15k's of extremely low Nash weight up to 1M. You may have noticed the prime 243163663*2^9190871 I have found recently. For the k's I'm testing usually only about 400500 n's survive the sieve and need to be tested by LLR. So this could be done on a typical machine in about one month or so. But the chances of finding a prime is at least very low. Nevertheless, if you find one it will be a big one
Until now I've computed Nash weights for all k<350.000.000 and there are a lot of interesting candidates, e.g. almost about 50.000 candidates have a Nash weight less than 100. For my tests I've picked only those with weights lower than 20. If you're interested in a list of possible candidates, e.g. the "less than 100 list", just leave a note here in the forum. Thomas L8 L19 L10 
20040306, 16:34  #3 
Sep 2002
2·131 Posts 
Hi Thomas,
Are you taking 1423*170881 (243163663) to 10M? This would definitely be a good candidate for team work. Feel free to post any of your work so I can add it to the stats. What program do you use to find nash weight? Joss 
20040306, 20:03  #4  
Jul 2003
wear a mask
649_{16} Posts 
Hi Thomas,
I would like to see that list; could a link to it be added to the 15k webpage? Regards, masser Quote:


20040307, 13:01  #5 
Feb 2003
2^{2}×3^{2}×53 Posts 
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 LLRtested 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=100.001110.000, while w' is the Nash weight obtained for n=110.000) 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 kvalues 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  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 
20040308, 04:21  #6 
Jun 2003
1,579 Posts 
Thomas,
Do you know which is the second smallest riesel number? Could you also provide me with a list for k's with weight's under 40. We should extend you list upto 2^31 than just 350M. Thanks, Citrix 
20040308, 10:48  #7 
Feb 2003
2^{2}·3^{2}·53 Posts 
Hi Citrix,
there are 5 Riesel numbers below k=1 million: 509203, 762701, 777149, 790841, and 992077. I'll send you a complete list of zero weighted k's as well as the "less than 40" list by private mail. Interesting to note, that there are much more k's of zero weight than k's with weights between 0 and 40 ... Currently I'm extending the weights list to k=500M, any coworkers are welcome, but the problem is how to store and exchange the large data blocks generated by the program (e.g. 10.5 MByte for a 1M krange). Well, one could cancel the w' computation and store k and w in binary form (4 + 2 Byte), but this is still 3 MByte instead of 10.5 MByte. Thomas 
20040308, 14:20  #8 
Jun 2003
1,579 Posts 
Thomas,
The algorithm used in psieve is inefficient. You can use recursion and reduce the number of run hours to less than 25. Citrix 
20040308, 20:33  #9 
Sep 2002
262_{10} Posts 
I'm working on the zero's up to k=2^31 the list is at the end of the stats. I'm at k=13M and testing them with newpgen one by one up to n=1M.
Joss 
20040309, 07:25  #10 
5^{2}·137 Posts 
Thomas,
are you 100% sure that a zero weight k is indeed a riesel prime? maybe the nrange for calculating the weight is just to low. I don't think so, but safty first. Thommy 
20040309, 10:13  #11  
Feb 2003
2^{2}×3^{2}×53 Posts 
Quote:
please do not mix up "Riesel primes" and "Riesel numbers"! A Riesel number is a k, for which k*2^n1 is never prime. You may verify the zero weight k's easily by hand and a pocket calculator. Let's take k=762701, for example: 3  762701*2^11 7  762701*2^21 3  762701*2^31 5  762701*2^41 3  762701*2^51 11  762701*2^61 3  762701*2^71 5  762701*2^81 3  762701*2^91 13  762701*2^101 3  762701*2^111 5  762701*2^121 3  762701*2^131 7  762701*2^141 3  762701*2^151 5  762701*2^161 3  762701*2^171 17  762701*2^181 3  762701*2^191 5  762701*2^201 3  762701*2^211 13  762701*2^221 ... The pattern repeats itself and shows that: 3 divides every second number, 5 divides every fourth number, 7 divides every sixth number, 11 divides every tenth number, 13 divides every twelth number, and 17 divides every sixteenth number. Or in other words: After sieving to p=3, only half the numbers remain (the even ones in the above case). Sieving to p=5 eliminates every second of the remaining numbers. p=7 should eliminate every sixth number, but half of them are already eliminated by p=5. And so forth ... It is special for Riesel numbers that the small primes up to 13, 17 (or some other small prime) cancel out ANY n. And therefore no primes can exist for that k. The above test may be done by NewPGen too  it should find for any given range of n, that after p=13, 17 (or,in some rare case, a higher "small" prime) no n remains. Thomas 
