20070413, 04:13  #12 
Jun 2003
Oxford, UK
1,901 Posts 
E58 Payam monsters
Some very high weight k from E58 payams:
32579969380008913005 8525 8521 (52 primes up to n=4000) 107251588234045704735 8406 8427 (51) 13807832114791617045 8422 8426 (55) 61105047309618807495 8435 8425 (59) 103580176519428233625 8419 8417 (59) 23414125704058421475 8406 8398 88808249896288517595 8396 8397 84315984863907089865 8387 8391 14785001407354998105 8402 8389 127417329985126803765 8388 8389 76394646230550311235 8390 8388 60906262455634018995 8384 8383 69653119293945259215 8383 8381 32553911156616844995 8371 8369 37287243950233762485 8356 8368 140901287463148857645 8362 8363 8510826865344068265 8367 8358 84981504576267664365 8356 8358 39007792020021273735 8358 8351 41461979133931229445 8366 8349 132699500417439184515 8345 8348 29875821764528915175 8331 8345 99215945779882385025 8342 8345 103449671808231170565 8342 8344 34392540738877805055 8337 8342 130261440899098442505 8360 8342 80147906108029855095 8335 8337 30351653111902964985 8335 8336 122887606923246617595 8335 8336 69489454966333401285 8344 8334 7386718138834077615 8329 8331 61665332067869044515 8334 8329 119444814610232214675 8327 8329 50246977722089253105 8340 8328 117259940292110939895 8325 8328 Again these are not necessarily rich in primes, that is all down to chance in the early n. None of the top 5 were rich in primes in the first 100 n. 
20070619, 23:39  #13 
Oct 2006
404_{8} Posts 
Where can I find this program that Axn1 produced?
The way I have found (some) k's is NewPGens Cunningham chain sieve. Thanks, Roger 
20070620, 01:18  #14  
Nov 2003
2·1,811 Posts 
Attached to post #9, a few posts up.
Quote:


20071216, 05:11  #15 
Oct 2006
2^{2}·5·13 Posts 
How are the nash weights calculated? Also, I've read above and elsewhere in mersenneforum that the higher the nash weight (like 8000+ as in the above posts), the better the chances of the k. However, I just found a reasonably good one with so far 112 by n=16332, and the weight is "3641 3652" In case it matters, it is a ep130 generated by payamx.
Thanks! roger Last fiddled with by roger on 20071216 at 05:14 
20071216, 08:13  #16 
"Curtis"
Feb 2005
Riverside, CA
11·401 Posts 
Nash weight is simply the number of candidates left after sieving to some arbitrary value (I think 512). The two numbers are from sieving the 010k range and the 100110k range.
Sieving to 1 million or some similar higher p would produce a better gauge of candidate density, but Nash is long established. The best "known" predictor of prime density is candidate density as far as we know, the chance a number has of being prime is a function of sieve depth, size of number, but not the actual kvalue. Some believe we will find other factors with more data, certain k's or types of k's more or less likely to produce primes than others, but nothing is yet known. I suppose one hope is a statistical analysis of large pools of data once we have such a pool; even then, determining causation is a long way from discovering patterns or potential patterns. Curtis 
20071216, 08:22  #17 
Mar 2006
Germany
5442_{8} Posts 
to calculate the nash weights look here: http://www.mersenneforum.org/showthread.php?t=7213.
higher nash is equal higher chance of finding is quite correct: the nash weight indicates how many candidates are remain after a sieve limit and therefore the more candidates the more primes are possible. it's only a point of reference not a guarantee! on the other side: the more candidates the more prime tests you have to do so it's very time consuming to go up in higher ranges of n! your k with nash 3641 and 112 primes you can see that the nash is only a value without a predictable number of primes of a k! (2nd info for rogers post, some minutes after ) Last fiddled with by kar_bon on 20071216 at 08:24 
20071216, 22:17  #18 
Oct 2006
2^{2}×5×13 Posts 
Thanks VBCurtis and kar_bon, it makes more sense now!
Regarding payam numbers, is there a website that details what e values have been searched, and to what level? I'm currently doing a search with ep130 (because I know the Riesel forum is for k*2^n1), have sieved for numbers to 1T, and tested them to 250B. So far I've had two k's give 99 primes by n=10,000, but none at the critical 100 mark Curtis, when you said 'candidate density' did you mean the average gap between primes up to a certain level (eg gap=100 by n=10,000 = 100 primes or gap=200 by n=10,000 = 50 primes)? Or the amount of primes below a certain value? Thanks! roger 
20071217, 00:24  #19 
Mar 2006
Germany
2×3×5^{2}×19 Posts 
the page from robert is here http://home2.btconnect.com/rwsmith/pp/payam1.htm but it's from 2002.
so the best info you can get from robert himself. see post #1 here. 
20071217, 01:46  #20 
Jun 2003
4,721 Posts 
He has an updated site at http://robert.smith44444.googlepages...umberresources

20071218, 01:11  #21  
"Curtis"
Feb 2005
Riverside, CA
4411_{10} Posts 
Quote:
You can think of the number of primes as a random variable, and look at its mean and variance, etc. Or, you can hope that past performance is a prediction of future prime density, and hope k's with high numbers of primes for their weight will continue to have higherthanpredicted numbers of primes going forward. Some searchers hope someday to have so much data that we can show this distribution is more than just noise that some k's really are more likely (or less) to produce primes. Curtis 

20071218, 18:37  #22  
Oct 2006
2^{2}·5·13 Posts 
Quote:
Quote:
Thanks! roger 

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