 mersenneforum.org program P-1 for K*2^n-1
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 Register FAQ Search Today's Posts Mark Forums Read 2003-12-28, 14:47 #12 jocelynl   Sep 2002 2×131 Posts Hi Val, I don't mind if you use it for any project. What is a General Mersenne Project? Joss  2003-12-28, 19:29 #13 Citrix   Jun 2003 1,579 Posts I have written a p-1 client . it is available here http://www.15k.org/********* There is alot of scope for improvements since the client does not use FFT or DWT and does GCD at every step.  Citrix   edit: Ill remove this from the server as this is slower than the current version available. Last fiddled with by Citrix on 2003-12-28 at 19:36  2003-12-29, 03:19 #14 Val   11×647 Posts Thanks Joss! Awesome, now I cant wait until the new LLR comes out. A Mersenne prime would be generally considered a Riesel prime of the form: (k*2^n-1) with n prime, and k=1. If you take all the Riesel primes (k<2^n), and list them in order of their size, R=3 7 11 23 31 47 79 127 191 223 239 n=2 3 2 3 5 4 4 7 6 5 4 R=383 479 607 863 1087 1151 1279... n=7 5 5 5 6 7 8 ... you will see that Mersenne primes, always have the largest exponent n, up that point. Hence the question, "what about the other largest exponents n?" This General sequence of primes includes all Mersenne primes, within it. The algorithm can be started from any known prime of the sequence, ie Mp. For instance starting with M7, the algorithm searches for the first prime, produced by an odd m: (k*2^n-1) + (m*2^n) = prime 127=(1*2^7-1) (1*2^7-1) + (1*2^7) = 255 composite (1*2^7-1) + (3*2^7) = 511 composite (1*2^7-1) + (5*2^7) =767 composite (1*2^7-1) + (7*2^7) =1023 composite (1*2^7-1) + (9*2^7)=1279 is prime 1279= (5*2*8-1) (5*2*8-1) + (1*2^8) = 1535 composite (5*2*8-1) + (3*2^8) = 2047 composite (5*2*8-1) + (5*2^8) = 2559 composite (5*2*8-1) + (7*2^8) = 3071 composite (5*2*8-1) + (9*2^8) = 3583 is prime 3583 = (7*2^9-1) repeat etc.. The bound of m, is unknown but, multiples of n, are used during searches. If a prime is not found within the first multiple of n, then search the next multiple, etc... As soon as the first prime is found, the algorithm is reset with the new exponent. There is some reason to suspect that when k is divisable by only, one and itself, and n is prime in this sequence, then k<=p. If this is true then it may be that all prime (k*2^p-1) with k<=p, are automatically a part of the sequence! Which makes them of the highest priority, since they create new points to start from, along with the already established Mersenne primes. Until a complete program is in place, the software used is: A. A script is used, to start a search, from a General Mersenne, or other(Thomas Ritschel/Shane Findley) B. This is sent to newpgen, and sieved.(Paul Jobling) C. Optionally, p-1 can be used to look for special factors.(Joss!) D. Then, the file is passed to LLR, or Prime95 for testing.(Jean/George) E. During the testing, a prime status button is now provided. (create shortcut to your desktop)(Thomas Ritschel F. Proth is used to double check positive results, with the SSE2 turned off. (Yves Gallot)  2003-12-29, 03:42 #15 Val   2,293 Posts P.S. We are testing exponent M 132049, for the next prime. The sequence is known from n=1-10000. Hopefully we can catch up to the top 5000 database, since M132049, is just short on digits. But all other Mersenne's are fair game, as of now. 2003-12-29, 05:01 #16 Citrix   Jun 2003 30538 Posts Val, Generalized Mersenne numbers seem intresting. Is there a website to join the search or to read more on the status or on the properties of these numbers. Citrix     2003-12-29, 23:03 #17 Val   22·137 Posts Citrix, Not quite yet. This started underground with group L9. But, Synergy = Energy! If there is enough interest to go around, then suggestions are welcomed. Poll?  2003-12-30, 11:42 #18 Val   791810 Posts Side note: It's pretty neat that each mersenne prime exponent, is actually how many time two apears as a factor, on the left side k, during the algorithms steps. ie, 1*2^7-1 Next primes until another Mersenne. 10*2^7-1 the same as 5*2^8-1 14*2^8-1 10*2^9-1 6*2^10-1 4*2^11-1 = 2^13-1 Take the original 7, and add the number of times two, is a factor of k, and we get 13 ding ding! Hey that is almost a Mersenne prime proof!  2003-12-31, 00:41   #19
Maybeso

Aug 2002
Portland, OR USA

2×137 Posts Quote:
 For instance starting with M7, the algorithm searches for the first prime, produced by an odd m: (k*2^n-1) + (m*2^n) = prime
Just curious: Each time you find a prime, how often is m > k? How often is m < k?
At first I thought when you get to a mersenne prime, (m = 1) < k.
And for the next one m > (k = 1). But I'm asking about k & m before you "normalize" them.

What would happen if you started at m = k instead of m = 1? Would you skip too many primes?
(like most of the mersenne primes?)

What would happen if you only checked m = 1..k and continued with the previous k,m pair if no primes found?

hmm, I guess that's not a good way to divide up the search.

oh, well. -- I really like your search sequence!  2004-01-11, 17:24 #20 Val   5·7·103 Posts This may clarify some thoughts on the sequence: Technically n, is how many times two, appears as a factor of an even k, during the algorithm. Those factors of two, are passed over to their proper, exponent index. For instance, if we take the expression of odd m above, and replace it with an even k. This is the form which the script, and LLR testing displays the k and n. k n = prime 4 0 = 3 2 2 = 7 4 3 = 31 4 5 = 127 10 7 = 1279 14 8 = 3583 10 9 = 5119 6 10 = 6143 4 11 = 8191 two is a factor of k, 13 times. The exponents of two, from k on the left, are added to n on the right, each prime found, p = 2+1+2+2+1+1+1+1+2 = 13 Expected approximations: K, stays relatively the same size as n. N, however on average, is twice as large as the number of general Mersenne primes found. This implies that Mersenne primes can be used to expect the next general Mersenne: Mp = 2^p-1 The next general Mersennne prime should be found by approximately this size: GMP = (p+2)*2^(p+2)-1 For example, if we look at the latest Mersenne prime M40, 2^20996011-1 (20996013)*2^(20996013)-1 The variable is usually small, since it has to follow with (p+2). Any GMP can be used to approximate the next one, using this method. This has been verified, so far up to M31. 1 2 M1 1 3 M2 1 5 M3 1 7 M4 5 8 1 13 M5 5 14 1 17 M6 1 19 M7 7 21 1 31 M8 5 32 1 61 M9 3 64 1 89 M10 3 94 1 107 M11 9 109 1 127 M12 95 128 1 521 M13 681 522 1 607 M14 113 608 1 1279 M15 1127 1280 1 2203 M16 3113 2204 1 2281 M17 143 2284 1 3217 M18 55 3221 1 4253 M19 741 4254 1 4423 M20 153 4426 1 9689 M21 1053 9690 1 9941 M22 103 9943 1 11213 M23 91 11217 1 19937 M24 769 19939 1 21701 M25 669 21705 1 23209 M26 7841 23210 1 44497 M27 6615 44499 1 86243 M28 1541 86246 1 110503 M29 1 132049 M30 86071 132051 25341 132053 85025 132054 1 216091 M31 62431 216093 ? The first woodall prime, that is also a general Mersenne prime is, 751*2^751-1 Factoring properties, of the algorithm's candidates: M19 PRIME 1048575 == 3 * 5 * 5 * 11 * 31 * 41__ 2097151 == 7 * 7 * 127 * 337 3145727 == 13 * 241979 4194303 == 3 * 23 * 89 * 683_______ 5242879 == 19 * 275941 6291455 == 5 * 1258291 7340031 == 3 * 3 * 3 * 271853______Every 3rd candidate is divisable by 3. 8388607 == 47 * 178481 9437183 == 7 * 37 * 83 * 439 10485759 == 3 * 3495253_________ 11534335 == 5 * 2306867 12582911 == 11 * 11 * 103991 13631487 == 3 * 61 * 74489_______ 14680063 G 7 21 PRIME In that same example, M19 PRIME 1048575 == 3 * 5 * 5 * 11 * 31 * 41__ 2097151 == 7 * 7 * 127 * 337 3145727 == 13 * 241979 4194303 == 3 * 23 * 89 * 683 5242879 == 19 * 275941 6291455 == 5 * 1258291___________Every 5th candidate is divisable by 5. 7340031 == 3 * 3 * 3 * 271853 8388607 == 47 * 178481 9437183 == 7 * 37 * 83 * 439 10485759 == 3 * 3495253 11534335 == 5 * 2306867__________ 12582911 == 11 * 11 * 103991 13631487 == 3 * 61 * 74489 14680063 G 7 21 PRIME This is true of any factor found, from it's first appearence onwards. Observations: There is some reason to suspect that when k is divisable by only, one and itself, and n is prime in this sequence, then it seems that so far k<=p. It would be nice to see, (k*2^p-1) prime, with the smallest k<=p, automatically in our sequence! This would make them a higher priority, since they create new points to start from for others. The advantage is, that there is'nt a restricting starting point, other than the smallest k<=p, producing a prime. Not to mention, the striking similarity to the Mersenne prime definition, and it's default nature. Heuristically from the 15k project, we had already found that when k, was prime it produced few primes n. By restricting n, to be prime as with Mersenne candidates, makes it an even more rare occurence. These could be considered a special Mersenne Prime, or SMp: k p 1 2 1 3 1 5 1 7 3 11 1 13 1 17 1 19 13 23 1 31 3 43 1 61 43 67 1 89 3 103 1 107 1 127 181 281 103 283 199 331 229 347 31 409 397 449 229 491 1 521 439 523 1 607 181 673 67 677 31 739 751 751 139 811 3 827 811 863 31 887 19 953 751 1033 67 1049 181 1091 541 1093 61 1103 241 1109 43 1171 421 1201 1 1279 ...?

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