Re: Mersenne: Generalized Mersenne Numbers
Brian J. Beesley <[EMAIL PROTECTED]> writes Congratulations on the (unverified) discovery of the 40th Mersenne Prime. I was thinking (always dangerous!) about generalizing Mersenne numbers. The obvious generalization a^n-1 is uninteresting because they're all composite whenever a>2 and n>1. However there is an interesting generalization: Define GM(a,b) = a^b-(a-1), so GM(2,b) = M(b); also GM(a,1) = 1 for all a The distribution of primes amongst GM(a,b) for small a > 2 and small b does seem to be interesting - some values of a seem to yield a "richer" sequence of primes than others. Note also that, in this generalization, some _composite_ exponents can yield primes. Another interesting point: the "generalized Mersenne numbers" seem to be relatively rich in numbers with a square in their factorizations - whereas Mersenne numbers proper are thought to be square free. (Or is that just Mersenne numbers with prime exponents?) A few interesting questions: (a) Is there a table of status of "generalized Mersenne numbers" anywhere? Some time ago I had a look at numbers of the form 2^n - 3, i.e. GM(4, n/2). Here are my results for 3320 <= n <= 16800: 2^n - 3 is a verified prime for n = 3954, 5630, 6756, 8770, 10572, 14114. 2^n - 3 is a probable prime for n = 14400, 16460, 16680. I don't know if someone else has verified the last three. Also 2^12819 - 7 (GM(8, 4273)) is a probable prime, 2^8824 - 15 (GM(16, 2206)) is a verified prime. The verified primes were done by factorization of N+1 and N-1, and APRCL. -- Tony _ Unsubscribe & list info -- http://www.ndatech.com/mersenne/signup.htm Mersenne Prime FAQ -- http://www.tasam.com/~lrwiman/FAQ-mers
Mersenne: The first 1000+ digit twin prime?
Thanks to all who responded to my 'what was the first 1000-digit prime' query. Another question: Can anyone tell me who discovered the first twin primes of 100+ digits, and when. I tried various Web pages and found 260497545*2^6625 +/- 1 (2003 digits, Atkin & Rickert, 1984), but none smaller or earlier. Thanks. -- Tony _ Unsubscribe & list info -- http://www.ndatech.com/mersenne/signup.htm Mersenne Prime FAQ -- http://www.tasam.com/~lrwiman/FAQ-mers
Mersenne: The first 1000+ digit prime
We all know that A. Hurwitz discovered the Mersenne primes 2^4253 - 1 and 2^4423 - 1 in 1961. (i) Were these the first two 1000+ digit primes discovered? (ii) If that is true, then is it generally accepted that the larger one (4423) was discovered first? (The story I heard was that left the computer running overnight and when he came to look at the results he read the printer output backwards, thus seeing 4423 before 4253.) -- Tony _ Unsubscribe & list info -- http://www.ndatech.com/mersenne/signup.htm Mersenne Prime FAQ -- http://www.tasam.com/~lrwiman/FAQ-mers
Re: Mersenne: Re: Pointers on farming
Steinar H. Gunderson <[EMAIL PROTECTED]> writes >>2. Which type of processor, memory etc will give the most bang for the buck? >> We rarely see anything beyond P2-250's, so I would have to pay retail for >>that. > Here is a calculation I did about six months ago AMD K6/2 500 MHz (plus fan)45 pounds Sterling Super-socket-7 motherboard 45 pounds 32M 100Mhz RAM 20 pounds Old 486 to put it in1 pound To this one should add the cost of electricity. The computer draws about 40 Watts. A Watt costs about 0.60 pounds per year. For, say, three years of computation, add 120 Watt-years of electricity 72 pounds -- Total 183 pounds That's 36.6 pence per MHz; or, since I am writing it off after three years, 0.389 pence per trillion cycles. -- Tony _ Unsubscribe & list info -- http://www.scruz.net/~luke/signup.htm Mersenne Prime FAQ -- http://www.tasam.com/~lrwiman/FAQ-mers
Re: Mersenne: The Second Mersennium Behind Us, How Now For MyriadThe Third?
Aaron Blosser <[EMAIL PROTECTED]> writes >Dunno 'bout all that, but another problem was that in order to do a "quick >and dirty" fix of the Y2K problem, a good number of people implemented >windowing. Some used a window of 1930-2029 (which most Microsoft software >uses to interpret 2 digit years), some used 1940-2039, etc. > >That gives those idiots another 29 years to fix the software the right way. One software company I know of is using a window of 1948-2047. So they could have a date problem in 2048. Surely this is the real 'Y2K bug'. -- Tony _ Unsubscribe & list info -- http://www.scruz.net/~luke/signup.htm Mersenne Prime FAQ -- http://www.tasam.com/~lrwiman/FAQ-mers
Mersenne: MM61
MM61. Thanks to all who joined in the search for a factor of 2^(2^61-1)-1. Keep up the good work! There is now a 'progress' page at http://www.ltkz.demon.co.uk/ar2/mm61.htm -- Tony _ Unsubscribe & list info -- http://www.scruz.net/~luke/signup.htm Mersenne Prime FAQ -- http://www.tasam.com/~lrwiman/FAQ-mers
Re: Mersenne: new (third) facotor of M(M(31))
Reto Keiser <[EMAIL PROTECTED]> writes >hi all > >i've found a new factor of M(M(31)) using Mfac 2.27. i am not >absolutely sure if it was already found by someone else, but with regard >to Will Edgingtons list, the factor isn't found yet. > >The factor is 242557615644693265201 ( 2*k*M31 +1 ) > >using a k of 56474845800( 112949691600 ( 2*k in Mfac )) > Congratulations and well done! It's certainly new to me. -- Tony _ Unsubscribe & list info -- http://www.scruz.net/~luke/signup.htm Mersenne Prime FAQ -- http://www.tasam.com/~lrwiman/FAQ-mers
Mersenne: MM61
If you have a PC and are interested in joining a coordinated search for a factor of the double-Mersenne number MM61 (= Mersenne number M2305843009213693951), have a look at http://www.ltkz.demon.co.uk/ar2/mm61.htm Thanks, -- Tony _ Unsubscribe & list info -- http://www.scruz.net/~luke/signup.htm Mersenne Prime FAQ -- http://www.tasam.com/~lrwiman/FAQ-mers
Re: Mersenne: Search for a divisor of M_M_61.
My previous announcement contained errors. Here it is again. You can download 'MFAC' from http://www.ltkz.demon.co.uk/ar2/mfac225.zip and e-mail me for a range. MFAC searches for divisors of double-Mersenne numbers M_M_e = 2^(2^e-1) - 1 for not-too-large exponents e. MFAC runs on any PC from a 486 upwards and on any system that supports MSDOS. You can even run it entirely from a bootable diskette. Memory usage is small and the files require less than a megabyte of disk space. The program is easily stopped and restarted. I have volunteered to coordinate a search for factors of M_M_61. Let d be a divisor of M_M_61. Then we know that d = N*(2^61 - 1) + 1. The parameters I intend to send out will set up MFAC to look for divisors of M_M_61 with N's in ranges of 204,204,000,000 To give some idea about timing, an AMD K6/2/400 will do a range in about 4 days. I intend to out ranges from N = 10^14 approx. to avoid clashing with work currently in progress by Sturle Sunde. -- Tony _ Unsubscribe & list info -- http://www.scruz.net/~luke/signup.htm Mersenne Prime FAQ -- http://www.tasam.com/~lrwiman/FAQ-mers
Mersenne: Search for a divisor of M_M_61.
After longer-than-expected delay my program is now ready. You can download 'MFAC' from http://www.ltkz.demon.co.uk/AR2/MFAC225.ZIP and e-mail me for a range. MFAC searches for divisors of double-Mersenne numbers M_M_e = 2^(2^e-1) - 1 for not-too-large exponents e. (It can also look for factors of Fermat numbers F_e = 2^2^e + 1.). MFAC runs on any PC from a 486 upwards and on any system that supports MSDOS. You can even run it entirely from a bootable diskette. Memory usage is small and the files require less than a megabyte of disk space. The program is easily stopped and restarted. I am volunteering to coordinate a search for factors of M_M_61. Let d be a divisor of M_M_61. Then we know that d = N*(2^61 - 1) + 1. The parameters I intend to send out will set up MFAC to look for divisors of M_M_61 with N's in ranges of 204,204,000,000 To give some idea about timing, an AMD K6/2/400 will do a range in about 4 days. According to Will Edgington, up to N=18,726,396,568 has been done. (Note: My N = Will's 2*k.) However it may be that someone is seriously continuing the search. Therefore I suggest, unless you specifically want to do smaller N's, that I start handing out ranges from, say, N = 10^13. One possibility is that we do all divisors with N from 10^13 to 10^14 and mop up 0 to 10^13 later. A word or two of warning. Unlike GIMPS, where we can be confident of finding a new Mersenne prime within a reasonable time, it may be the case that we are attempting the impossible. All we can do is hope that (i) M_M_61 is composite, and (ii) it has a factor small enough to be discoverable. -- Tony _ Unsubscribe & list info -- http://www.scruz.net/~luke/signup.htm Mersenne Prime FAQ -- http://www.tasam.com/~lrwiman/FAQ-mers
Re: Mersenne: Important info on M(M(p)) from Wilfrid Keller
Lucas Wiman <[EMAIL PROTECTED]> writes >[...] >> when I found out than Curt Noll had started an attack on M(M(127)) with >> superior hardware. I imagine that by now he has carried the search much >> further. > >This is further evidence for why GIMPS is such a good idea! (as if we >needed more) Well, yes ... but in practice when you work on a problem with considerably more powerful hardware than has previously been applied to it, there is little cost in starting again from scratch, and it double- checks the work that has already been done. >If Curt Noll and you have been working together, your computer time would >not have been wasted, nor would (presumably) much of his. This is the >beauty of distributed computing projects, but this illistrates how *key* >centrilization is. Of course, soon after Curt started working on M_M_127 I collaborated with him - by retiring! >Now, I think that it might be a nifty (tm) idea if GIMPS/primenet >were to start looking for factors of double Mersenne numbers. If I >understand correctly, much of the code is already written (changes to >the P-1 code, and the normal factoring code), and I'm sure that such things >could interest mathematicians more than finding the next Mersenne prime. The only feasible approach for M_M_61 and above is trial-division. I recently blew the dust off by my program MFAC and gave it a run. For M_M_61 it processes divisors N*M_61 + 1 on an AMD/400 at about 2.2 billion N's per hour. (6 billion/hour for M_M_31.) If people are interested, and unless someone else can do better, after a bit of tidying up I can make this program available. Then we need a mechanism for handing out work. I am happy to volunteer, but it will have to be done manually. -- Tony _ Unsubscribe & list info -- http://www.scruz.net/~luke/signup.htm Mersenne Prime FAQ -- http://www.tasam.com/~lrwiman/FAQ-mers
Re: Mersenne: Important info on M(M(p)) from Wilfrid Keller
>From: Wilfrid Keller <[EMAIL PROTECTED]> >I had found in 1976 might suggest, I have shown interest in this >particular kind of Mersenne numbers since many, many years. Re- >garding the factor of M(M(31)) just rediscovered by Lucas Wiman, >I regret to inform you that it was already "published" in 1983. >Unfortunately, Guy Haworth's notes (please see the references be- >low) were released only "privately", but were in fact widely cir- >culated. The second prime factor of M(M(31)), found by Tony >Forbes, was also known to me for some years (again, see below). I must admit that at the time I found it, I suspected this second factor of M(M(31)) might not be new; it seemed too easy. >Search limits h < L(p) for factors of >"iterated" Mersenne numbers M(M(p)) as of November 1996 > >pL(p) > > 127 6.8 x 10^8 I was working on M(M(127)) about 2 years ago. I completed ranges of h (for possible divisors 2*h*M(127)+1) 0 - 12972960 36036000 - 39039000 39039000 - 42042000 45045000 - 48048000 and was part way through ranges 12972960 - 24024000 24024000 - 36036000 42042000 - 45045000 48048000 - 54054000 when I found out than Curt Noll had started an attack on M(M(127)) with superior hardware. I imagine that by now he has carried the search much further. -- Tony _ Unsubscribe & list info -- http://www.scruz.net/~luke/signup.htm Mersenne Prime FAQ -- http://www.tasam.com/~lrwiman/FAQ-mers
Re: Mersenne: STOP BASHING PEOPLE WITH SLOWER MACHINES!
"Steinar H. Gunderson" <[EMAIL PROTECTED]> writes >On Thu, Jul 29, 1999 at 01:31:13PM -0600, Aaron Blosser wrote: >>Egads...I'm not telling *anyone* to NOT use slow computers...I'm >>"suggesting" they use < p166 for double-checks or factoring. ... >>I hope I'm being clear that I don't consider slower computers useless >>entirely...just useless for doing LL tests in the range of exponents >>currently being assigned. >Not useless, just slow. I still think it makes perfect sense. I remember discussing a while back the problem of undetected errors. I came to the conclusion that, for example, the probability of getting a 600-ish exponent wrong is about 1/20. I assumed that the error rate is proportional to number of cycles in the LL test. Then someone suggested that it is more realistic for the error rate to be proportional to the duration of the run. It could be a serious problem for slow computers that take months and months to do a single LL test. An undetected bit error in the wrong place would invalidate the entire run. On the above assumption the chance of this happening is inversely proportional to the speed of the computer. For slow computers it could therfore be considerably worse than the 1/20 I estimated. On the other hand, factoring is relatively immune because there is not the long-range interdependence that exists with the LL test. Also the job of factoring is to find factors. There are no 'negative' results to collect. It is not a disaster if the occasional factor is missed. However, it was also pointed out that old processors and memory are more reliable. -- Tony _ Unsubscribe & list info -- http://www.scruz.net/~luke/signup.htm Mersenne Prime FAQ -- http://www.tasam.com/~lrwiman/FAQ-mers
Re: Mersenne: Pepin's Test, etc.
"Brian J. Beesley" <[EMAIL PROTECTED]> writes >In case anyone else notices that F25 is an "interesting" size - >10,100,891 digits - could I save you a whole bundle of time by >remarking that F25, F26 and F27 are all known to be composite, we >know at least one factor of each of them. The cofactor of F25 (after dividing out all the known factors) is currently 'status unknown', isn't it?. So it might be worth testing for compositeness. However at present there seems to be no way of proving it prime if it fails the compositeness test. -- Tony _ Unsubscribe & list info -- http://www.scruz.net/~luke/signup.htm Mersenne Prime FAQ -- http://www.tasam.com/~lrwiman/FAQ-mers
Re: Mersenne: STOP BASHING PEOPLE WITH SLOWER MACHINES!
Sturle Sunde <[EMAIL PROTECTED]> writes >I use almost only relatively slow computers. More than 100 of them. If >they are only 1/5 as fast as what some people think is apropiate, it means >that they can do 100 tests in the same time as an appropiate machine does >5. While a very few people think I should stop using those computers for >testing, I'm no 23 on the GIMPS (not Primenet) top-100 list. (I can't >find this Mr. "Go Away Loosers With Slow Computers" there, but he is >probably working under some pseudonym.) The problem I have with slow computers is that I cannot afford to run them. A 40MHz 486 uses about 20 Watts, a 400MHz AMDK6 about 40 Watts. However if you have your own hydroelectric power supply, or if someone else is paying the electricity bill, the situation is different. -- Tony _ Unsubscribe & list info -- http://www.scruz.net/~luke/signup.htm Mersenne Prime FAQ -- http://www.tasam.com/~lrwiman/FAQ-mers
Re: Mersenne: RE: Mersenne Digest V1 #568 & prize question
Paul Leyland <[EMAIL PROTECTED]> writes > >> Paul Leyland <[EMAIL PROTECTED]> wrote: >> >> >> a new prize pool of $.11 has been >> >> proposed (the radix is 10 :-) If you think it's worth a few >> > >> >The radix is always 10. I guess you mean the radix is >> (1+1)*(1+1+1+1+1), >> >> That made me laugh! Good point... >> >> >or, more concisely, (1+1+1)(1+1) + 1. >> > >> >Can anyone represent that number in fewer than (1+1+1)! ones? >> >> Yes (but difficult to write in plain ASCII): >> >> (1+1+1+1) >> - >> \ >> \i >> / >> / >> - >>i = 1 >> >> or: Sum(i=(1) to (1+1+1+1)) over i. >> >> Cornelius Caesar :-) > >But that's cheating! You're allowed only 1 as a numeric quantity, possibly >modified with any of +-*/! and sqrt. > How about [sqrt([sqrt(sqrt((1+1+1)!!))]!)], where the square brackets [x] denotes, as usual, the integer part of x. -- Tony Unsubscribe & list info -- http://www.scruz.net/~luke/signup.htm
Re: Mersenne: usefulness and 486's
"David M. Moore" <[EMAIL PROTECTED]> writes >I now have this 486 machine doing the factoring that was >being done by the P120. Generally, it is taking almost two weeks per >exponent. You could upgrade to a P120 very cheaply. In England the going rate for a P120 CPU is about 20.00GBP and 1.00GBP for a suitable motherboard. In the US I should imagine it's the same numbers in $s. People are almost giving them away. >Therefore, my question is even with this two week time, is the 486 >machine doing "useful work" for GIMPS, or is it merely heating up the >CPU? I ask this because in a couple or three weeks I may have access >to a quantity (30 to 75) of 486DX-50 machines. If these machines can >contribute useful work to GIMPS, I will happily give them each a mouthful >of exponents to factor :-) It's a nice thought, but you really need cheap electricity to make it work. I found that a typical 486DX-33 (I don't have a -50 to hand) uses about 21 Watts, and that's without an HDD. Multiply by 75 and it could start to get expensive, although in the winter months you would make a saving on your heating costs. By contrast an AMD/K6/2-400 on a basic motherboard (no sound, no video) and no HDD draws only about 35 Watts. -- Tony Unsubscribe & list info -- http://www.scruz.net/~luke/signup.htm
Mersenne: factorization
I know it's a bit feeble the usual standards for ECM factorization, but I just found this 36-digit factor of 2^2203+1 : 848425589476255002200418022118728331 Thus 2^2203+1 = 3 * 13219 * 208613913329 * 743372438374143806443 * 282324958084937237369 * 848425589476255002200418022118728331 * C571 I apologise if this is not new or not interesting. I am still trying to find numbers n > 1000 where gcd(n, 30) = 1 and 2^n- 1 and 2^n+1 are completely factorized. Like 2203 would be once the C571 part has been factorized (2^2203-1 is prime). The only ones I know are 1121 and 1141. Anyone know of any more? -- Tony Unsubscribe & list info -- http://www.scruz.net/~luke/signup.htm
Re: Mersenne: Double-checking (was Chronons)
George Woltman <[EMAIL PROTECTED]> writes >Finding an error in the first LL test is not rare. I've said about 1 in >200 are incorrect. When the entire 1,400,000 - 2,000,000 range has >been double-checked I'll perform some more rigorous analysis of the >reliability of first-time LL results. This is slightly worrying. The number of CPU cycles for performing the LL-test for exponent n is approximately proportional to n^2*log(n). Assuming, maybe rather naively, that the risk of computer error is proportional to the number of CPU cycles, the LL-test for a typical exponent of around 600 could be about 9.7 times less reliable than for one in the vicinity of 200. However, modern Pentium-IIs and memory chips are probably much more reliable than the 486's and ordinary Pentiums that did the LL-tests in the range 140 to 200. On the other hand, an error rate of 1/20 is just about acceptable. Do we have any statistics for the larger exponents? -- Tony Unsubscribe & list info -- http://www.scruz.net/~luke/signup.htm
Mersenne: Completely factorized (2^n-1)(2^n+1)
Can anyone help? I am looking for odd numbers n (not necessarily prime), n not divisible by 3, n > 1000 such that both 2^n-1 and 2^n+1 have been completely factorized into primes. The Cunningham tables give 1121 and I have been unable to find any more. However, there has been a lot of recent factorizing activity and my sources may well be out of date. I am also interested to know who found the factors 58142098448088409 and 359071640268582342735956401 of 2^1121+1. Thanks, -- Tony Forbes
