Alkulukuja tms. Sekalaisia hauskoja nyyssipostauksia ja linkkivinkkejä ------- Subject: Top-10 prime numbers http://www.glasgowg43.freeserve.co.uk/pfrecs.htm ------- Math and numbers (plenty): http://www.greenhodge.net/g/read/math/ http://www.greenhodge.net/g/inc/text.php?filename=read/math/is-this-a-prime.txt&pagetitle=is-this-a-prime ------- "Verficification at http://news.bbc.co.uk/hi/english/sci/tech/newsid_1693000/1693364.stm I especially liked Woltman's statement that there were still more primes to find." ------- "Harry J. Smith" wrote: > > I updated my "Plot of Mersenne Primes." > > see: > > http://home.netcom.com/~hjsmith/Perfect/MersPlot.html > http://home.netcom.com/~hjsmith/Perfect/MerDPlot.html > http://home.netcom.com/~hjsmith/Perfect/Mersenne.html Nice plots, Harry. The first is saying to me "another one isn't far away" :-) (maybe I should take my pills!). Is the 'index' of the previous one really unknown - as you have a question mark next to the index. (It seems they've all had at least one LL test in that range.) Phil ------- >The Ulam spiral is created by the procedure: > 12 11 10 9 > 13 2 1 8 > 14 3 k 7 > 15 4 5 6 > 16 ... >where k can start as any integer. The longest CONTINUOUS prime line >is >40 primes, produced when k=41, corresponding to Euler's n^2 + n + 41 >for n=[0,39], with discriminant -163 (which is Class no 1). >I was playing around with D. Morin's excellent applet at >http://platon.lacitec.on.ca/~dmorin/applet/village (I must have too >much time on my hands ) and wondered what was the SECOND >longest >line. >It seems to be 23 primes, when k=3527, corresponding to 4n^2 - 2n + >3527 for n=[-2,20], with discriminant -14107, class no. 11. >True? Looks good!! Along with the quadratics!! Is there an actual proof that Euler's n^2 + n + 41 is the largest prime producer of 40 primes? Or just that no other equation has been found to produce > 40 primes. Dan ------- Of quadratics of the form, an^2 + an + p where p is a prime, then YES, they've proven that Euler's polynomial is the best, since it is the only one of that form with discriminant -163, the largest number with class number 1. (I read that in the old mathworld.wolfram.com, which now has a Chinese domain name.) You can, of course, have quadratics of the form, an^2 + bn + p The best of these was found by Ruby, 36n^2 -810n + 2753 which generates 45 primes, from n=[0,44]. BUT, some of those are negative, so you have to be generous with your definition of primes. See: http://www.primepuzzles.net/problems/prob_012.htm Not all quadratics are straight lines in Ulam's spiral. For example, 2n^2 + 29, which produces 28 consecutive primes. No matter what k you choose, you can't arrange the 28 primes in a single line. I'm interested in Ulam spiral quadratics. A line of consecutive primes (5 or more) in the Ulam spiral will have a quadratic form with a discriminant that is always (I think) one of the numbers of the list of class numbers. For a partial listing of those class numbers, you can try the "Online Enclyclopedia of Integer Sequences": http://www.research.att.com/~njas/sequences/ (By the way, the Ulam spiral quadratic I mentioned, 4n^2 - 2n + 3527, has a discriminant -14107 which is the THIRD largest with class number 11. The website I cited didn't give the complete list.) --Tito Piezas III ------- Date: Thu, 29 Mar 2001 14:42:29 GMT NNTP-Posting-Host: 131.228.71.83 This is the 'press release' sent to the PrimeNumbers and NMBRTHRY mailing lists, but I thought there might be some interest here. Phil ------ Four months ago, to the day, David Underbakke was pleased to announce[1] the discovery of the largest known twin primes[2], 665551035*2^80025+/-1, with 24099 digits. The above was achieved after searching less than the expected search-space, so, feeling encouraged by that, we decided to collaborate again to attack a higher target. The target has finally been reached: at 29603 digits each, 1807318575 * 2^98305 +/- 1 are twin primes. One of the lessons learnt from the former search was the importance of thorough pre-sieving of the ranges to remove all candidates with small (<10^14 or similar magnitude) prime factors. Pre-sieving reduced the number of lengthy probable- primality tests required. This is doubly effective when searching specifically for twin primes, as the density improvement #twins/PrP-test is squared (e.g. sieving further to increase the single-prime PrP density by 10% increases the twin density by 21%). After some discussion, we came to the conclusion that we could get the most from the sieving stage by again looking at primes of the form K.2^N+/-1, and by chosing an exponent N optimised for the sieving stage. Phil had in the past used this technique, and we decided to simply use the same exponent, N = 98305 = 0x18001. As before, a dedicated twin sieve was used, using Phil's own code on his Alpha 21164 (which was hand-optimised specifically for this exponent) and using Paul Jobling's NewPGen[3], a general-purpose pre-siever. An arbitrary odd number of this size (assuming the mean K tested is of order K=10^9) is prime with probability P(single) = 2/ln(10^9.2^98305) = 1/34080 Therefore the probability of finding a twin prime was P(Twin) = P(single)^2 * 1.32 = 1/880M where the adjustment factor 1.32 is twice the twin prime constant[4] (and M is million). Therefore an odd K range of 1 .. 2.09G (i.e 1045M candidates) could be expected to yield one twin. (In retrospect, it appears our failure potential was a bit high perhaps!) We sieved for about 2GHz-months on several machines, until we had sieved up to p=50T. Using Merten's theorem[5], we can predict that a simple sieve to p=50T increases the density for arbitrary numbers by 1.781*ln(50*10^12) = 56.2, but a twin sieve increases the density by 56.2^2/1.32 = 2390. However, we were not checking even numbers anyway, by the construction of our expression, so the real density increases ought to be 1:28.1 and 1:1195, respectively. This predicts a candidate count within .1% of what we ended up with - 875000 candidates. After that it was 'simply' a matter of distributing the work across the various machines. The probable primality tests were done using George Woltmann's PRP program. In order to give Phil's Alpha something to do once the sieving was complete, we opted to do the +1 test first (for the Alpha to prove formally) and only perform the -1 test on PrPs from the +1 test. As hinted at before, the bulk of the work began about four months ago (Phil started the sieve in advance). David was able to put between 10 and 15GHz onto the project on average, and Phil about half that. During the search we found many hundred solo primes, but a few weeks ago, as we started handing out the final blocks to the various PCs, it looked as if the twin search would be fruitless. However, yesterday, with perfect thriller timing, PRP provided a positive result to the -1 test on one of David's machines. It was a very quick matter to then prove the -1 case formally[6] using Yves Gallot's Proth, and the +1 case[7] using Phil's own code. Our thanks go to the authors of the software mentioned above and to Professor Caldwell for the Prime Pages website, without which we'd not have been equipped to make our predictions (and thus know where to aim). David Underbakke, Phil Carmody - 28/03/2001 References: [1] Underbakke; Largest Twin Primes Record Broken, Nov 28 2000 http://listserv.nodak.edu/scripts/wa.exe?A2=ind0011&L=nmbrthry&F=&S=&P=2877 All of the following are from Professor C. Caldwell's paedagogical Prime Pages, and are given relative to and can be navigated to from both http://primepages.org/ and http://www.utm.edu/research/primes/ [2] Twin Primes. /lists/top20/twin.html [3] NewPGen. /programs/NewPGen/index.html [4] Twin Prime Constant /glossary/TwinPrimeConstant.html [5] Merten's Theorem /glossary/MertensTheorem.html [6] n-1 tests and Pepin's Test For Fermats. /prove/prove3_1.html [7] n+1 tests and the Lucas Lehmer Test. /prove/prove3_2.html ------ A google search for 'NFSnet' yielded: http://orca.st.usm.edu/~cwcurry/nfs/nfs.html which has source. And a world famous factorer recently pointed me towards FactorWorld, which might be useful, for example: http://www.maths.usyd.edu.au:8000/u/contini/factoring/FactorCode.html Phil -------