Mersenne Digest Friday, December 5 2003 Volume 01 : Number 1095
---------------------------------------------------------------------- Date: Tue, 02 Dec 2003 17:50:09 -0600 From: [EMAIL PROTECTED] Subject: Re: Mersenne: 40th Mersenne Prime verified and word is getting out > >I'll try to add more links to the http://mersenne.org web page as they become >available. I visit news.google.com fairly often. On their main page just now, they had several articles on this. I'm sure a search would reveal more as they come out. http://zdnet.com.com/2100-1103_2-5112827.html http://www.theinquirer.net/?article=12985 (Claims GIMPS is a "peer to peer" system. Does that mean ya'll can send me music?) http://washingtontimes.com/upi-breaking/20031202-013854-9481r.htm http://www.newscientist.com/news/news.jsp?id=ns99994438 - - Stephen Whitis - --- www.whitis.com _________________________________________________________________________ Unsubscribe & list info -- http://www.ndatech.com/mersenne/signup.htm Mersenne Prime FAQ -- http://www.tasam.com/~lrwiman/FAQ-mers ------------------------------ Date: Thu, 04 Dec 2003 07:05:51 +0100 From: "Jean Penné" <[EMAIL PROTECTED]> Subject: Re: Mersenne: 40th Mersenne prime verified ; Generalized Mersenne numbers Hi All, Involved in GIMPS since 1996, I am very happy to congratulate Michael, George, Scott and all other participants in this great project, for this magnificent success! Also, entering this adventure gave me the desire for the hunting of large primes, and the wish to create or improve software tools helping to do that. I wish also to answer to Brian J. Beesley about "Generalized Mersenne numbers". To test the primality of a^b-(a-1) numbers, I know no algorithm based on Lucas Sequences... But I did some work on 3^n-2 numbers, which interested my friend J.J Kessis, Professor at Paris University. I sieved for n=2 to n=1,000,000 using Newpgen -> 78500 remaining n's after 24H. Next, I found 29 pseudoprimes for n up to 60928, using PRP. the last found is n=37056 Last, I certified the primality, using Marcel Martin's Primo, for : n = 22, 37, 41, 90, 102, 105, 317, 520, 541, 561, 648, 780, 786, 957, 1353, 2224, 2521. (n = 2, 4, 5, 6, 9 yield also prime numbers...) The last certified prime has 1203 digits, and the test duration is 2H48'38" on a 2.5Ghz P4 I thank that these results were too small to be transmitted, but perhaps I was wrong... Also, I thank that numbers of the form (b^n-1)/(b-1) were better candidates for the name "Generalized Mersenne numbers", because their divisibility properties are exactly the same (all divisors are k*n+1), but they are also called "Generalized repunits" and, unfortunately, no efficient special algorithm seems to be known to test them... Regards, Jean Penné _________________________________________________________________________ Unsubscribe & list info -- http://www.ndatech.com/mersenne/signup.htm Mersenne Prime FAQ -- http://www.tasam.com/~lrwiman/FAQ-mers ------------------------------ Date: Thu, 4 Dec 2003 00:42:36 -0800 (PST) From: Gordon Irlam <[EMAIL PROTECTED]> Subject: Mersenne: Off topic - historical computing manuals I know there are few people interested in computing history on this list. Would anyone be interested in any of the following historical manuals: Control Data 6400/6500/6600 Computer Systems Reference Manual, 1969 This was the world's first commercial supercomputer and first RISC architecture based machine; designed by Seymour Cray. PDP 11 bus and processor handbooks, ~1973 VAX 11/780 hardware, software, and architecture handbooks, ~1977 Bell System Technical Journal, 1978, Vol 57, No 6, Part 2 This is a special edition devoted to the Unix Time-Sharing System authored by all of the key players; it might be the first major publication exploring Unix in detail, I am not sure. Bell Laboratories Technical Journal, 1984, 63, 8, 2 Another full issue devoted to Unix by key players. The Oak Language and Oak Virtual Machine Specification Release 0.9, 1994 Oak was subsequently renamed Java. I checked with the Computer History musuem (a great place if you have never been), and they didn't want them, but I am having a hard time just throwing them away. Let me know if you are interested. Just give me your address, and I will mail them to you. thanks, gordon _________________________________________________________________________ Unsubscribe & list info -- http://www.ndatech.com/mersenne/signup.htm Mersenne Prime FAQ -- http://www.tasam.com/~lrwiman/FAQ-mers ------------------------------ Date: Thu, 4 Dec 2003 11:22:25 +0100 (MET) From: Wojciech Florek <[EMAIL PROTECTED]> Subject: Mersenne: Generalized mersennes Brian Beesley wrote: > Date: Sun, 23 Nov 2003 08:53:55 +0000 > From: "Brian J. Beesley" <[EMAIL PROTECTED]> > Subject: Mersenne: Generalized Mersenne Numbers > Define GM(a,b) = a^b-(a-1), so GM(2,b) = M(b); also GM(a,1) = 1 for all a What about GM(p,n)=p^n-(p-1)^p For p=2 we have GM(2,n)=2^n-1^2=M(n) For a prime p>2 we have GM(p,mp)=p^mp-(p-1)^p= (p^m)^p-(p-1)^p so it is divisible by p^m-(p-1) (for m=1 it equals 1). I have no idea how to prove (if it is possible to prove) that GM(p,n) is composite for composite n. However for p=3 we have (p-1)^p=2^3=8, so n should be greater than 1 and GM(3,2)=9-8= 1 (not prime, but not composite, too) GM(3,3)=27-8=19 prime GM(3,4)=81-8=73 prime GM(3,5)= 235= 5*47 GM(3,6)= 721=7*103 (as should be, see above) GM(3,7)= 2179 prime GM(3,8)= 6553 prime (for composite exponent!) GM(3,9) divisible by 3^3-2=25 GM(3,10), GM(3,11) composite GM(3,12) divisible by 3^4-2=79 GM(3,13) composite For p=5 we start from n=5 since 4^5=1024, whereas 5^4=625 GM(5,n) composite for n=5,6,8,9,10 (div. by 25-4=21) for n=7 and 11 -- prime GM(5,7)=77101 GM(5,11)=48827101 What happens if p is composite? [for p>3 we always start from n=p since in this case p^p > (p-1)^p and p^{p-1}<(p-1)^p, p=3 is the exception] GM(4,n) is composite for n<12 GM(6,11) is prime! 6^11-5^6=362797056-15625=362781431 Maybe it is caused by fact that p-1=5 is prime? What about more general meresennes MGM(p,q,n)=p^n-q^p, p,q prime (or not)? q^p is related to small Fermat theorem (I don't remember exactly how; probably q^(p-1) mod p =1 for prime p). As regards the analog of the LL test, I only remember that Lucas series is somehow related to Fibonacci series (see D.Knuth and others "Concrete mathematics"). Is similar series are related with a^b-(a-1) or a^b-(a-1)^a? Regards W Florek PS The most important: CONGRATULATIONS to Micheal, George and Scott and many, many others. WsF =============================================== Wojciech Florek (WsF) Adam Mickiewicz University, Faculty of Physics ul. Umultowska 85, 61-614 Poznan, Poland phone: (++48-61) 8295033 fax: (++48-61) 8295167 email: [EMAIL PROTECTED] _________________________________________________________________________ Unsubscribe & list info -- http://www.ndatech.com/mersenne/signup.htm Mersenne Prime FAQ -- http://www.tasam.com/~lrwiman/FAQ-mers ------------------------------ Date: Fri, 5 Dec 2003 17:05:30 +0100 From: "Paolo Fani" <[EMAIL PROTECTED]> Subject: Mersenne: Re: 40th Mersenne Prime verified > Michael Shafer discovered the 40th known Mersenne prime, 2^20996011-1. Am I too late to congratulate with Michael, George, Scott and you all for this result? Very well done! Sincerely, Paolo p.fani @ infogroup.it p.s. as an amusement, have a look at the digit distribution of such prime, compared with that of an equal amount of decimals from gamma, e and pi. Last column is delta with expected average for a random distribution. 2^20996011-1 0: 631705 9,994652% -338 1: 632720 10,010711% +677 2: 630989 9,983324% -1054 3: 631467 9,990887% -576 4: 632004 9,999383% -39 5: 633283 10,019619% +1240 6: 630929 9,982375% -1114 7: 633503 10,023100% +1460 8: 632964 10,014572% +921 9: 630866 9,981378% -1177 -------- 6320430 mean: 632043 ± 1008,880 = 632043 ± 0,159622% (n-1) mean: 632043 ± 957,108 = 632043 ± 0,151431% (n) - -------------------------------------------------- gamma(6320430) 0: 631692 9,994447% -351 1: 631788 9,995965% -255 2: 632064 10,000332% +21 3: 632594 10,008718% +551 4: 632435 10,006202% +392 5: 633459 10,022404% +1416 6: 631379 9,989494% -664 7: 631295 9,988165% -748 8: 631961 9,998703% -82 9: 631763 9,995570% -280 -------- 6320430 mean: 632043 ± 644,335 = 632043 ± 0,101945% (n-1) mean: 632043 ± 611,270 = 632043 ± 0,096713% (n) - -------------------------------------------------- e(6320430) 0: 631149 9,985855% -894 1: 633197 10,018258% +1154 2: 630780 9,980017% -1263 3: 633443 10,022150% +1400 4: 632955 10,014429% +912 5: 632294 10,003971% +251 6: 631789 9,995981% -254 7: 632132 10,001408% +89 8: 630789 9,980160% -1254 9: 631902 9,997769% -141 -------- 6320430 mean: 632043 ± 957,178 = 632043 ± 0,151442% (n-1) mean: 632043 ± 908,058 = 632043 ± 0,143670% (n) - -------------------------------------------------- pi(6320430) 0: 631536 9,991978% -507 1: 632438 10,006250% +395 2: 631748 9,995333% -295 3: 631940 9,998370% -103 4: 632319 10,004367% +276 5: 631942 9,998402% -101 6: 630422 9,974353% -1621 7: 632985 10,014904% +942 8: 632241 10,003133% +198 9: 632859 10,012911% +816 -------- 6320430 mean: 632043 ± 731,600 = 632043 ± 0,115752% (n-1) mean: 632043 ± 694,057 = 632043 ± 0,109812% (n) - -------------------------------------------------- Finally, of course, the biggest perfect (delta rounded to the nearest integer) (2^20996011-1)*2^20996010 0: 1265474 10,010982% +1388 1: 1263035 9,991687% -1051 2: 1263218 9,993135% -868 3: 1265979 10,014977% +1893 4: 1264223 10,001085% +137 5: 1262192 9,985018% -1894 6: 1264569 10,003823% +483 7: 1263798 9,997723% -288 8: 1263617 9,996291% -469 9: 1264753 10,005278% +667 -------- 12640858 mean: 1264085,8 ± 1152,539 = 1264085,8 ± 0,0911757% (n-1) mean: 1264085,8 ± 1093,394 = 1264085,8 ± 0,0864968% (n) _________________________________________________________________________ Unsubscribe & list info -- http://www.ndatech.com/mersenne/signup.htm Mersenne Prime FAQ -- http://www.tasam.com/~lrwiman/FAQ-mers ------------------------------ End of Mersenne Digest V1 #1095 *******************************
