Mersenne Digest Tuesday, May 25 1999 Volume 01 : Number 563 ---------------------------------------------------------------------- Date: Fri, 21 May 1999 14:24:22 -0600 From: "Aaron Blosser" <[EMAIL PROTECTED]> Subject: RE: Mersenne: Odd observation > Which brings me to my odd observation... Could the cooler CPU > temp speed up > operations by that much? Or am I going insane? I know that the > physics of it > says it should be a *little* bit faster, but .02s/iteration faster? > > Someone please explain this to me. :) > > Jeremy Well my brother, I figure that the clock chip is the culprit. Perhaps the case fan you had before wasn't properly cooling the insides enough, and clock chips are sensitive to heat. There's really no reason why the CPU would do better if the clock speed were a constant, so I can only conclude that it's the clock itself that is running faster. Go team MadPoo! :-) Hanging steady at 40th place, but watch out...I got some new dual PII-450 machines! ________________________________________________________________ Unsubscribe & list info -- http://www.scruz.net/~luke/signup.htm ------------------------------ Date: Sun, 23 May 1999 06:26:38 -0500 From: kilfoyle <[EMAIL PROTECTED]> Subject: Mersenne: bogus ID assignments > I just found that I still had an system generated ID on one of my > systems from back on March 13 . The S06036 should be combined with > MKilfoyle and then s06036 can be deleted. When I corrected the ID and > PW on PC RED the new data is now listed but old resultrs are still > under the bad ID. I also saw several other IDs in the list of a a > similar naming series. Is it possible that we have other PC out there > doing work under the wrong ID for other people?? Michael.. > ________________________________________________________________ Unsubscribe & list info -- http://www.scruz.net/~luke/signup.htm ------------------------------ Date: Tue, 25 May 1999 19:41:20 +0100 From: [EMAIL PROTECTED] Subject: Re: Mersenne: Repeating LL remainders [EMAIL PROTECTED] writes: > We can illustrate working modulo M23 = 8388607 = 47 * 178481. >3 is a quadratic reside modulo 47 (e.g, 12^2 == 3 mod 47), >so 2 + sqrt(3) is in the base field of order 47. >The multiplicative order of 2 + sqrt(3) divides 47-1, and turns out to be 23. > > 3 is a quadratic nonresidue modulo q = 178481. The order of 2 + sqrt(3) >divides q + 1 = 178482 = 2 * 3 * 151 * 197 and turns out to be 178482. > > The least common multiple of these orders is >2 * 3 * 23 * 151 * 197. So L(2 * 3 * 23 * 151 * 197) == L(0) mod M23. > > For the L sequence, we need two powers of 2 whose difference is >divisible by 2 * 3 * 23 * 151 * 197. >The orders of 2 modulo 3, 23, 151, 197 are 2, 11, 15, 196, respectively. >The order of 2 modulo 3 * 23 * 151 * 197 is the least common multiple >of these orders, namely 2^2 * 3 * 5 * 7^2 * 11 = 32340. >To include a factor of 2, we need L(2^32341) == L(2^1) mod M23. >That is, S(32341) == S(1) mod M23. >This is far more than 23 LL steps before the sequence repeats. > > EXERCISE: Repeat this analysis modulo M11 = 23 * 89. > Find the period modulo M29 = 233 * 1103 * 2089, after getting > the individual periods for each prime factor. Yes - the result for M29 is quite interesting due to the "unexpected" short period of 60. (I feel after a week I'm not spoiling it for anyone by giving the result away...) Incidentally the first repeating remainder for _all_ the non-prime Mersenne numbers with prime exponents is S(1) = 14. I was interested to see what would happen for larger exponents, in particular if the first repeating remainder would always be 14. Unfortunately this is not true, the first repeating remainder for M37 is S(5) = S(516929). Finding this took a fair bit of CPU time as, to start with, I more or less assumed that the remainder would eventually go back to 14 (Peter's impeccable logic seemed to suggest this to my inadequate maths!) so I had to run through 2^37-1 calculations to be sure that 14 never recurred, and even then I wasn't sure I wasn't fighting a program bug. Now I have a _much_ faster way of checking, assuming that the first repeating remainder will be one of the first few in the (S) sequence. But is this true? Instinctively one feels that it ought to be, but can one construct an example where the the index n of the first repeating remainder S(n) is large compared with the period? My math is totally inadequate to investigate this... (I'm only interested in the case p prime, M(p) composite. Of course when M(p) is prime then n=p and the period is 1) Also, Peter's logic seems to depend on knowing the factors. Would it be possible to "reverse engineer" at least one of the factors of M(p) (or at least be in a much-improved position to find them) if one knew the repeat period and/or the first repeating remainder in the S sequence for M(p)? The reason I ask is that my "experimental" method of finding the period & remainder does not depend on _any_ knowledge of the factorization of M(p), and I reckon I could get the values for, say, M727 with the expenditure of a great deal less computer time than completing the search to B1=44*10^6 using ECM would take. Regards Brian Beesley ________________________________________________________________ Unsubscribe & list info -- http://www.scruz.net/~luke/signup.htm ------------------------------ End of Mersenne Digest V1 #563 ******************************
