20070422, 14:05  #1 
Jun 2003
Oxford, UK
3·5^{4} Posts 
A new very prime k
Working as I do with nice large k, I thought I would introduce you to k=531131527270075522241760982081252274580435
I don't know what the nash weight is here, because k is too large for the nash calculator I have. This k looks to as if it might better the currently known ultimate in producing primes as it is a Payam 162 number (no k in the power series k*2^n1 has a prime factor where the order base 2 of primes is 162 or less), and has many many more small primes than the average such k. n=19763 and 19764 provide the largest of 7 Sophie Germains, 6 of which are over n=100, and 13521 and 13523 produced a near miss. I will soon complete testing to n=100,000. At the time of writing I have tested to 90,137 and it has produced 142 primes, well up there with the best of k's discovered in my + search, and because of the density of numbers requiring checking can be expected to produce many more up to 1,000,000. The current record, plus or minus, is 169 primes for a k, on the plus side. 
20070424, 17:16  #2  
Jun 2003
Oxford, UK
1875_{10} Posts 
Quote:
Progress, now with this big k is 143 primes and n=101317, and I only had 7 plus numbers better than this. Record for 143 primes...n=76635. all of these numbers had smaller nash weights. 

20070426, 13:24  #3  
Feb 2003
11·173 Posts 
Quote:
(That's quite close to the max. possible Nash weight of 10000). A modified version of the Nash calculator (which now supports your ultra large values of k) is attached (source and Linux binary). 

20070426, 13:41  #4 
Nov 2003
E26_{16} Posts 
A 42digit k! And quite an attractive one. Shall we try team sieving? But reaching 1M is going to be a major task and will take us ages

20070426, 13:49  #5 
May 2005
2·809 Posts 
Yeah, I am having troubles reaching n=700000 with my 9digit k=736320585 with ~142 candidates in every 1000 range of n

20070426, 15:11  #6 
Feb 2003
11×173 Posts 
Note, that Geoff's srsieves are restricted to k<2^64 (or 2^63?). Only Phil's ksieve would be able to cope with a k like this.
Perhaps, one should consider a multiple k team sieve on a small set of about 10 k or so, which are within the capabilities of Geoff's srsieves (up to about 20 digits). The problem with those high weight ks is that you'll get lots of primes at the smaller ns, but you'll encounter large gaps once you reach the "interesting" regions of n>250000 or so (means: primes which would enter the Top5000 list). Taking a multiple k set would give some nice averaging over these gaps (as we already learned from our other team drives). But means even more LLR tests to reach n=1M, of course... Last fiddled with by Thomas11 on 20070426 at 15:12 
20070426, 16:37  #7  
Jun 2003
Oxford, UK
3×5^{4} Posts 
Quote:
If there is a way to split the Newpgen into segments for distributed computing, it might work. In the meanwhile, the following are prime for n above 100000 101060 102124 106385 So now I have 145 primes, and have checked as far as 111185, and I have my second machine now, working on 130000  140000 Next week I will inherit two big desktops so I am hoping to start to generate some real work, instead of theorising and playing around with small primes. 

20070426, 16:46  #8 
Jun 2003
Oxford, UK
3·5^{4} Posts 
If you can create a windoze version I can check the 800 or so payam 162 numbers I have generated, and see if there is a larger Nash weight  either that or I can send you the file of numbers.

20070426, 16:56  #9  
Jun 2003
Oxford, UK
3·5^{4} Posts 
Quote:
BTW, the most prime k through the 100000 to 250000 range produced: 104186 105283 107911 122428 123760 125392 126478 129152 129850 137749 172113 174521 184521 202474 206409 214727 226714 231290 240057 243709 That is a lot of primes, and where they come up is anyones guess. PS. this prime had a gap from 243709 to the recently discovered 300332. So in one way you are right. We could prove you wrong 

20070426, 17:28  #10  
Feb 2003
11101101111_{2} Posts 
Quote:
A Windows binary of the Nash calculator is attached, though I don't know whether it works on a nonP4 (nonSSE2) machine... Last fiddled with by Thomas11 on 20070426 at 17:29 

20070429, 08:19  #11 
Jun 2003
2^{4}×97 Posts 
Robert,
I am interested in finding k's that produce alot of primes. How did you come up with this k? Please explain your method. Thanks! 
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