[QUOTE=Cruelty;108963]BTW: sometime ago I have sieved n=1234567 for k<20bit till p=82T. I am wondering what is the porbability that there is a prime among the remaining candidates (18283)?[/QUOTE]

About 70%. Expected # of primes: 1.2

[QUOTE=axn1;108988]About 70%. Expected # of primes: 1.2[/QUOTE]How do one get those numbers? :unsure:

[QUOTE=Cruelty;108991]How do one get those numbers? :unsure:[/QUOTE]

Probability of a random number N being prime = 1/ln(N). For k*2^1234567+/-1, you can approximate it to 1/(1234567*ln(2)).

Probability after sieving upto p = (e^gamma*ln(p))/ln(N)

Here, gamma is Euler's constant. You can take e^gamma = 1.781.

So, the probability of one of your candidates to be prime = (1.781*ln(82T))/(1234567 * ln(2)) = 57.06/855736 = 1/14997 = 6.67e-5

The prob that /none/ of your 18283 surviving candidates is prime is (1-1/14997)^18283 = 29.5%. So, prob that at least one prime is present is 70.5%

Expected number of primes = 18283 * 1/14997 = 1.22

Thanks! :smile:

Probability of prime after sieve, nicely stated!
 

Nicely stated and calculated, axn1! The only part that really lost me was the e^gamma part. Sounds like part of a calucation for the area under the bell-shaped curve. How was the e^gamma constant arrived at?

I enjoy the math and have done a fair amount of statistical analysis but I'm no higher-math whiz so if the exaplanation involves too much calculus differenciation or integration, then it may be above my head. But I figured I'd ask anyway.

Thanks for the analogy on the chances of a number being prime after being sieved to a certain amount. I'll use it a lot in the future to decide how to divide up work on my machines.


Gary

Thanks, axn, for the formula. Very much fun to "predict" the number of primes left in a candidate pool.
-Curtis

reserving k=2145
 

I've freed up a machine from twin-prime searching so I'm going to reserve another, most interesting, k, that is k=2145, to fill a gap and then continue from the highest found prime.

Being the 'base' factor for many high-weight k's, I'm amazed that it has not been tested more than it has. It has a gap from n=50K to 115899 and there is no indication that it has been tested higher than its highest prime of n=214994.


Gary

Gary-
2145 has 7 primes from 50k to 160k, the limit of my current search. A few months ago I got interested in k>2000, and began working on 2055, 2085, 2115, 2145, 2175. I have them LLR'ed to 160k so far, with the sieve to n=600k approaching p=4T.

I neglected to reserve them for no good reason-- you'll find a thread well down the list about k>1000 started by me; I began sieving back then, and just never posted the reservation. I guess I thought it unimportant until I got above top-5000 cutoff, but we didn't have anyone like you working on small numbers back then.

I can send you the primes and sieve if you'd like to work higher; you can just tell me your new limit when you tire of the work, and I'll pick it back up from there. You're welcome to work as high as you please. I have 160-210k ready for work now, 210+ still sieving. Please, allow me to send you at least this chunk!

-Curtis
p.s. reserving the other 4 now, please- 2055, 2085, 2115, 2175.

Sharing the glory on k=2145
 

I sure am good at stepping all over others people k's! :wink: I guess I shouldn't be surprised that someone was working on this one at some point...sorry about that.

It's really just been in the last 2 weeks that I've looked at lower k's with gaps. For the first 6-7 weeks of my prime-finding existence, I was looking for very high-weight k's with gaps or that just looked extremely good to me. But I just decided that I wanted to fill all gaps, regardless of the size of k, and it is the lower k's that are more visible.

I'm aware of 4 of the 7 primes that you found between n=50K and 160K because they are posted on the summary site. I'm guessing that Thomas previously found them. Since you have the entire range searched with all of its primes, I'd just suggest posting the primes to fill the gap in the 'small primes found' forum and Karsten can post them all. There's no reason to send them to me since you found them and I don't see a reason for me to double-check them since we would all know for sure that you've tested the entire range.

That's a heck of a sieve file! This is a bit awkward. Tell you what...I'd really like a crack at taking this one higher if your OK with providing me your sieved file. If I find a top-5000 prime, we can share in the glory. I think that's the way it usually works on the big projects...that is the coordinator, top siever, top searcher, and the finder all get credit for the find. Personally, it's doesn't make much difference to me. As you know the glory to me is just having complete and accurate lists of primes but I wouldn't mind having a prime or two in the top-5000 just for grins. :smile: I will just start from the last prime found of n=214994.

The sieved file to n=600K to P=4T :surprised is probably more than I'd be willing to test but I'll have to see what my machine situation is in the next couple of months. I have a friend who's going to lend me an 850-Mhz machine tomorrow for several months so that will help. I'll use it for some of my 'side projects' like twin prime searches that don't need high-speed processors but that have been taking time away from my speedy machines at different times lately.

Oh, and speaking of side projects...I have the ultimate gap-filling file that I will be sending on to Karsten / Kosmaj in the next couple of days. I'll create a separate thread for it. You'll probably want to see it when I post it. Karsten, I'm going to keep you really busy! But I'll be nice and warn you a little while ahead of time. :smile:


Thanks a bunch.


Gary

There is no need to share in the glory- if you find a 2145 prime big enough for the top 5000, it's all yours. You have done plenty of work for our project, and deserve some fame and fortune.

PM me your email, and I'll send the the sieve from where I stopped (162k or so) to 240k; n>240k is still sieving, so I'll send the rest when you get up to 240k. My trusty old celeron566@850 sieves at just over a trillion a month on this file.

-Curtis

gd_barnes:
"... Karsten, I'm going to keep you really busy! But I'll be nice and warn you a little while ahead of time. "

oh. no problem! i'm hunger for new informations to put them in the summary pages, to fill gaps and kill errors!!!
keep our summary for all k*2^n-1 the best, actual and completest on net is my purpose!
i'll keep an eye on this forum (and quite many other pages like Top5000) every day and all new info i obtain i put in the summary.
so sending your ultimate gap-filling file is like christmas and easter at one day for me :grin: