MM61 Налаштування

[nikelong]

MM61
http://anthony.d.forbes.googlepages.com/mm61.htm

 

http://www.rechenkraft.net/wiki/index.php?title=MM61

 

 

MM61
A search for a factor of 2261 − 1 − 1

It is not known whether the double-Mersenne number MM61 = 2261 − 1 − 1 is prime or composite, and, just as with other Mersenne numbers, it is interesting to resolve this question one way or another.

There is no hope of testing MM61 for primality. At 694127911065419642 digits it is far too large for the usual Lucas-Lehmer test. However, if MM61 is composite (and it very probably is), it might have a factor which is small enough to be discoverable. This project is a coordinated search for such a factor.

If you have a PC and are interested in helping to find a factor of 2261 − 1 − 1: First download the program 'MFAC' and perform a simple test to see that the software works on your computer. Then e-mail me for a range of numbers to test.

Email address: anthony.d.forbes@gmail.com

MSDOS and Windows: Get the MSDOS version of MFAC from here.

MFAC is designed to run under DOS using a proprietary 'DOS extender'. It should run quite happily alongside other applications, except that some people have experienced problems with Windows XP. Please try the thing out thouroughly and alongside other applications before volunteering.
Background

Let d be a prime divisor of MM61. We know that

d = N (261 − 1) + 1

for some N congruent to 0 or 2 (mod 8). We use a sieving process. We start with a long list of Ns and we reject all those Ns for which the corresponding d is divisible by a prime q = 3, 5, 7, 11, ... up to some suitable limit. For each N that survives we compute d and see if it divides into MM61. For this test we do not have to operate on MM61 itself. All calculations are done modulo d. We start with x = 2 and compute x2 (mod d) sixty-one times to obtain 2261 (mod d). If the final result is 2, then d divides MM61.

Of course, MM61 is not the only double-Mersenne number. The first four, MM2 = 7, MM3 = 127, MM5 = 2147483647, and MM7 = 170141183460469231731687303715884105727, are prime. The next four, MM13, MM17, MM19 and MM31 have known factors. And that's the extent of our knowledge. All other double-Mersenne numbers are 'status unknown', and MM61 happens to be the smallest. A number of people (including the author of this page) have tried finding divisors of MM61, MM89, MM107, MM127, MM521, MM607 and others, but so far without success.
MFAC

Program MFAC searches for divisors of double-Mersenne numbers, 22e − 1 − 1, for not-too-large exponents e. It can also look for factors of Fermat numbers, 22e + 1.

It runs on any PC from a 486 upwards and on any system that supports MSDOS. Memory usage is reasonably small, the files require less than a megabyte of disk space and the program is easily stopped and restarted. MFAC seems particularly well suited to AMD CPUs.

The parameters I am sending out will set up MFAC to look for divisors N (261 − 1) + 1 of MM61 with N in ranges of 204,204,000,000. A 400 MHz AMD K6/2 will do a range in about four days.
Further Information

PROGRESS

General information about prime numbers: The Largest Known Primes

Mersenne numbers: The Great Internet Mersenne Prime Search

Double-Mersenne numbers: Will Edgington's pages

Fermat numbers: Wilfrid Keller's page

***

Tony Forbes 19 February 2005.

Це повідомлення відредагував nikelong: Mar 9 2009, 13:43

[Death]

A search for a factor of 2^(2^61 − 1) − 1

It is not known whether the double-Mersenne number MM61 = 2^(2^61 − 1) − 1 is prime or composite, and, just as with other Mersenne numbers, it is interesting to resolve this question one way or another.

There is no hope of testing MM61 for primality. At 694127911065419642 digits it is far too large for the usual Lucas-Lehmer test. However, if MM61 is composite (and it very probably is), it might have a factor which is small enough to be discoverable. This project is a coordinated search for such a factor.

If you have a PC and are interested in helping to find a factor of 2261 − 1 − 1: First download the program 'MFAC' and perform a simple test to see that the software works on your computer. Then e-mail me for a range of numbers to test.

Хотят найти делитель для 2^(2^61 − 1) − 1.

Прогресс http://anthony.d.forbes.googlepages.com/mm61prog.htm

Ranges 0 to 19999 reserved: 16627 done, 3373 to do.