User group for PFGW & PrimeForm programs Yahoo Group Riesel/Sierpinski in base 3 =============================================== Guido Smetrijns Message 1 of 43 May 24, 2004 ----------------------------------------------- Andrew, and all others, It seems to me like if you're looking out for Riesel/Sierpinski numbers in the base 3. [k*3^n+/-1, k even of course, odd k's are trivial R/S numbers in odd bases]. You mention having found at least one prime for every even k up to 2 million, in both the plus and minus case. Great job, I like it! But I think you might be in for a long journey here... From a different point of view I recently did some elementary searching myself . As an example the smallest Riesel number (I only looked at the minus-case) found so far was the 27-digit number 169397207256533310164792176. Maybe I can push this limit down one or two digits relatively easy, but at present I think it will be quite hard to find an even R/S-number of less than 20 digits, say. Unless I'm overlooking something trivial of course. Furthermore I don't have the appropriate tools (yet) or the time to go for an in-depth search but maybe some other guys out there can help... Anyone ? Kind regards, Guido Smetrijns. Leuven - Belgium. ----- Original Message ----- From: andrew_j_walker To: primeform@yahoogroups.com Sent: Friday, May 21, 2004 2:58 AM Subject: [primeform] Looping until a prime is found In pfgw, is there a way to use an ABC file or otherwise, so that when testing k*b^n +- 1, when a prime is found that k value is stopped and a new one started? I've been doing some work on and off on k*3^n +- 1 and finding the first prime for a given k value. At the moment I'm running a pari script to find k values with no prime up to n=1000, outputting these to a file and then testing further with proth.exe I'd like to do this with pfgw so that a) It may run faster b) Tests are deterministic rather than probable in the n<=1000 stage c) More automated. With part c for instance, I'd like to be able to pick an n range of say 1001-10000 and loop over all remaining k values without having to manually stop and start for each one! So far my results for both the plus and minus case have found primes for all k up to 2 million, but this needs double checking for typos, composites being declared prime, etc. before I place the list somewhere. Thanks, Andrew ============================================= Guido Smetrijns Message 2 of 43 May 25, 2004 ----------------------------------------------- Hi Andrew, and all others, It seems to me like if you're looking for Riesel/Sierpinski numbers in the base 3. [k*3^n+/-1, k even of course, odd k's are trivial R/S numbers in odd bases]. You mention having found at least one prime for every even k up to 2 million, in both the plus and minus case. Great job, I like it! But I think you might be in for a long journey here... From a different point of view I recently did some elementary searching myself . As an example the smallest Riesel number (I only looked at the minus-case) found so far was the 27-digit number k=169397207256533310164792176, which is way beyond the "size" of the base-2 R/S-problem. Maybe I can push this limit down one or two digits relatively easy, but at present I think it will be quite hard to find an even R/S-number of less than 20 digits, say. Unless I'm overlooking something trivial of course. Furthermore I don't have the appropriate tools (yet) or the time to go for an in-depth search but maybe some other guys out there can help... Anyone ? Kind regards, Guido Smetrijns. Leuven - Belgium. ----- Original Message ----- From: andrew_j_walker To: primeform@yahoogroups.com Sent: Friday, May 21, 2004 2:58 AM Subject: [primeform] Looping until a prime is found In pfgw, is there a way to use an ABC file or otherwise, so that when testing k*b^n +- 1, when a prime is found that k value is stopped and a new one started? I've been doing some work on and off on k*3^n +- 1 and finding the first prime for a given k value. At the moment I'm running a pari script to find k values with no prime up to n=1000, outputting these to a file and then testing further with proth.exe I'd like to do this with pfgw so that a) It may run faster b) Tests are deterministic rather than probable in the n<=1000 stage c) More automated. With part c for instance, I'd like to be able to pick an n range of say 1001-10000 and loop over all remaining k values without having to manually stop and start for each one! So far my results for both the plus and minus case have found primes for all k up to 2 million, but this needs double checking for typos, composites being declared prime, etc. before I place the list somewhere. Thanks, Andrew =============================================== Shane Message 3 of 43 May 25, 2004 ----------------------------------------------- You should try an algorithm like the Riesel Mersenne Algorithm. It would be generalized to use base 3, rather than base 2. In this case the algorithm passes the number of times k is divisable by 3, over to the base exponent, every time a prime is found. k*3^n-1 For instance lets start with k and n = 1 reset algorithm by finding the number of times k is divisable by 3. 1*3^1-1=2 Prime!reset alg 1*3^1-1 same since k is not divisable by 3. 3*3^1-1=8 6*3^1-1=17 Prime! reset alg 2*3^2-1 3*3^2-1=26 6*3^2-1=55 9*3^2-1=80 12*3^2-1=107 Prime! reset alg 4*3^3-1 6*3^3-1=161 9*3^3-1=242 12*3^3-1=323 15*3^3-1=404 18*3^3-1=485 21*3^3-1=566 24*3^3-1=647 Prime! reset agl 8*3^4-1 ... and so on. These primes always have the largerest exponent, when classified in order of their size(digit length). Compared of course to other Base 3 Riesel/Sierpinski primes. =============================================== andrew_j_walker Message 4 of 43 May 25, 2004 ----------------------------------------------- Hi Guido, I started this search as a bit of fun and out of curiosity. What I've found out is that the highly composite k values are a lot sparser, that's why I've been able to get up to 2 million. Perhaps someday in the future someone will be able to apply a new method and reduce your example significantly. Have you tried any other bases? Andrew --- In primeform@yahoogroups.com, "Guido Smetrijns" <guido.smetrijns@c...> wrote: > Andrew, and all others, > > It seems to me like if you're looking out for Riesel/Sierpinski numbers in the base 3. [k*3^n+/-1, k even of course, odd k's are trivial R/S numbers in odd bases]. > You mention having found at least one prime for every even k up to 2 million, in both the plus and minus case. Great job, I like it! But I think you might be in for a long journey here... > From a different point of view I recently did some elementary searching myself . As an example the smallest Riesel number (I only looked at the minus-case) found so far was the 27-digit number 169397207256533310164792176. > Maybe I can push this limit down one or two digits relatively easy, but at present I think it will be quite hard to find an even R/S- number of less than 20 digits, say. Unless I'm overlooking something trivial of course. Furthermore I don't have the appropriate tools (yet) or the time to go for an in-depth search but maybe some other guys out there can help... Anyone ? > > Kind regards, > Guido Smetrijns. > Leuven - Belgium. =============================================== andrew_j_walker Message 5 of 43 May 25, 2004 ----------------------------------------------- Shane, can you please send me a reference for this algorithm? I'm not clear as to what it is doing and how it works. Andrew =============================================== Shane Message 6 of 43 May 25, 2004 ----------------------------------------------- There no other know references, as far as I know. Here is what I have found about the algorithm in base two: A Mersenne prime would be generally considered a Riesel prime of the form: (k*2^n-1) with n prime, and a fixed k = 1. Definition: If you take all the Riesel primes (k<2^n), and list them in order of their size,(digit length) R=3 7 11 23 31 47 79 127 191 223 239 383 479 607 863 1087 1151 1279... n=2 3 2 3 5 4 4 7 6 5 4 7 5 5 5 6 7 8 ... you will see that Mersenne primes by default nature, always have the largest exponent n, up that particular point. Hence the natural question, "what about the other largest n?" Is there a distinguishing property? The answer is yes! Algorithm: 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 next 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 next 3583 = (7*2^9-1) repeat etc.. Mersenne numbers, and primes, that have already been double checked by (G.I.M.P.S.) need'nt be tested. The maximum bound of m is some unknown but, multiples of n are used as an initial searching field. If a prime is not found within the first multiple/s of n, then the next multiple is searched, etc. Though if you plan to continue to test for the next prime afterwards, multiple fields of n are your best option. As soon as the first prime is found, the algorithm is reset with the new exponent. Technically n, is how many times two, appears as a factor of k, for each prime found 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 algorithm) 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, p = 2+1+2+2+1+1+1+1+2 = 13 This is true of any candidate's exponent, gM or Mn for that matter. Distinguishing property: Regular Riesel primes, such as 11,23,47,79,191,223,239,383,... cannot nessesarily be used as starting points, to find the largest exponent n. For instance from 191, the next prime found using it's k, and exponent, is 383. Although, they are still in an increasing order, and eventually merge back into the general sequence, if you start from them. The algorithm also works similarly with Proth primes of the form k*2^n+1. So one could also call this program RMPFA, or Riesel Mersenne Proth Fermat Algorithm. This will be investigated later as a Robinson project k*2^n+/-1, when enough data surrounding Mersenne primes is gathered. Regular Riesel's are much like tree branches, that merge into the stem, or master sequence. Expected approximations: Both k&n stay relatively the same size. However n, is about twice as large, as the number of general Mersenne primes found, although there should be some logarithmic explanation involving e^gamma. This directly implies that Mersenne primes, can be used to estimate the next general Mersenne. Mp = 2^p-1___gM = (p+2)*2^(p+2)-1 For example: 2^20996011-1___(20996013)*2^(20996013)-1 The variable is usually fairly small(though at it's largest after Mp), since it has to follow with (p+2). Any gM can be used to approximate the next one, using this method. This has been verified so far, up to M31. 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 gM 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 gM 7 21 PRIME This is true of any factor found, from it's first appearence onwards. Once a prime is found, the remainder of the sieved newpgen file, can be used to help sieve the next gM, since the algorithm overlaps some of it's candidates. (See "Salvaging old Newpgen files" below) Estimation: With k<2^n, the probability, that (k*2^n-1) is prime should be about 1/log(k*2^n-1). From a visual standpoint, Mersenne primes, tend to be on the lower side of the logarithmic wave, like steps. We see that after every Mersenne, there is a tendency to have a very large gap, between it and the next gM. Visuals of Mersenne steps: 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 25415 110504 1 132049 M30 86071 132051 25341 132053 85025 132054 1 216091 M31 62431 216093 97065 216095 Program to find these primes in base two found here, RMA.RAR http://15k.org/rma/ Group here: http://mersenneforum.org/forumdisplay.php?f=16 =============================================== andrew_j_walker Message 7 of 43 Jun 2, 2004 ----------------------------------------------- Unfortuneately a few computer/power outages have slowed my run down but hopefully in the next few days I'll be able to complete it. For fun I extracted a list of the first 10000 prps found to see how pfgw would go at proving them prime. Using -f -tp pfgw-prime.log had 9921 entries and pfgw.log had none. Checking the log file I found some entries of the following type: Primality testing 1588*3^13-1 [N-1/N+1, Brillhart-Lehmer-Selfridge] factors: 2531784923 However, testing without -f could prove this prime. Is this related to the bug Jim previously mentioned? There were several more examples of this, the next for 1808*3^13-1 Andrew =============================================== jim_fougeron Message 8 of 43 Jun 3, 2004 ----------------------------------------------- --- In primeform@yahoogroups.com, "andrew_j_walker" <ajw01@u...> wrote: > > Unfortuneately a few computer/power outages have slowed my run > down but hopefully in the next few days I'll be able to complete it. > For fun I extracted a list of the first 10000 prps found to > see how pfgw would go at proving them prime. Using -f -tp > pfgw-prime.log had 9921 entries and pfgw.log had none. Checking the > log file I found some entries of the following type: > > Primality testing 1588*3^13-1 [N-1/N+1, Brillhart-Lehmer-Selfridge] > factors: 2531784923 > > However, testing without -f could prove this prime. Is this related > to the bug Jim previously mentioned? There were several more examples > of this, the next for 1808*3^13-1 This is a "similar" but different bug. I will add that to the to get fixed. > Andrew =============================================== jim_fougeron Message 9 of 43 Jun 3, 2004 ----------------------------------------------- This bug (and others) have been fixed, and a updated development version of PFGW has been placed into the OpenPFGW yahoo groups' file folder. I will not place the Win32 EXE files into this group (primeform), until this build has time for some people to use it. However, if I hear of no complaints, I will put that version into the primeform file folder in a few weeks. Jim. --- In primeform@yahoogroups.com, "andrew_j_walker" <ajw01@u...> wrote: > > Unfortuneately a few computer/power outages have slowed my run > down but hopefully in the next few days I'll be able to complete it. > For fun I extracted a list of the first 10000 prps found to > see how pfgw would go at proving them prime. Using -f -tp > pfgw-prime.log had 9921 entries and pfgw.log had none. Checking the > log file I found some entries of the following type: > > Primality testing 1588*3^13-1 [N-1/N+1, Brillhart-Lehmer-Selfridge] > factors: 2531784923 > > However, testing without -f could prove this prime. Is this related > to the bug Jim previously mentioned? There were several more examples > of this, the next for 1808*3^13-1 > > Andrew =============================================== andrew_j_walker Message 10 of 43 Jun 8, 2004 ----------------------------------------------- Thanks for the updates Jim. I've now also finished the k*3^n+1 run up to n=10 million and k=8191. The only difference this time is I added a few more fermat test bases and factoring to 1000 if it passes these tests (not before!!). This adds a little to the testing time, but should significantly decrease the number of composites getting through. I'm now running the composite file (3.33 million entries!) through the newest winpfgw using -f -tp. I'll report any it has problems with here. Andrew =============================================== andrew_j_walker Message 11 of 43 Jun 16, 2004 ----------------------------------------------- Jim and all, I finally managed to get the winpfgw run completed which I mentioned below. The good news is that with the extra bases used and the trial factoring no pseudo-primes slipped through. The bad news is that I found 23 numbers that were missing from pfgw-prime.log , i.e. pfgw -f -tp returned as composite. I checked these with Dario Alpern's factoring applet and they all were found to be true primes. Note that the powers are all clustered together (except for two with n=7) so it may be an FFT length problem. However they are all declared as PRPs so maybe not. I also noticed for the first few that checking without the -f will prove them prime. The machine that produced this problem was a P4. I checked on an older machine (3) and it didn't have this problem. Here's an example where the proving doesn't work: C:\ajw\base3>pfgw -f -i -tp -q208496*3^39-1 PFGW Version 20040608.Win_Dev (Beta 'caveat utilitor') [FFT v23 w/P4] ***** Using functions from the static linked GMP library CPU Information (From Woltman v22 library code) Intel(R) Pentium(R) 4 CPU 1.50GHz CPU speed: 1513.57 MHz CPU features: RDTSC, CMOV, PREFETCH, MMX, SSE, SSE2 L1 cache size: 8 KB L2 cache size: 256 KB L1 cache line size: 64 bytes L2 cache line size: 64 bytes TLBS: 64 Primality testing 208496*3^39-1 [N+1, Brillhart-Lehmer-Selfridge] trial factoring to 65536 Running N+1 test using discriminant 29, base 1+sqrt(29) Running N+1 test using discriminant 29, base 2+sqrt(29) 208496*3^39-1 is composite (0.0216s+0.0017s) Andrew 208496*3^39-1 1711312*3^40-1 2039008*3^42-1 2234744*3^33-1 2235880*3^39-1 3677336*3^35-1 4738784*3^33-1 5281352*3^36-1 5564912*3^32-1 6372964*3^7-1 6648752*3^40-1 6797008*3^34-1 7160248*3^30-1 7552672*3^28-1 8435168*3^34-1 8454136*3^35-1 8911112*3^36-1 8921680*3^34-1 8967794*3^7-1 9217396*3^31-1 9497560*3^38-1 9510824*3^33-1 9521204*3^33-1 =============================================== jim_fougeron Message 12 of 43 Jun 16, 2004 ----------------------------------------------- --- In primeform@yahoogroups.com, "andrew_j_walker" <ajw01@u...> wrote: > > Jim and all, > > I finally managed to get the winpfgw run completed which I mentioned > below. The good news is that with the extra bases used and the trial > factoring no pseudo-primes slipped through. The bad news is that I > found 23 numbers that were missing from pfgw-prime.log , i.e. > pfgw -f -tp returned as composite. I checked these with > Dario Alpern's factoring applet and they all were found to be true > primes. Note that the powers are all clustered together (except for > two with n=7) so it may be an FFT length problem. However they are > all declared as PRPs so maybe not. I also noticed for the first few > that checking without the -f will prove them prime. > > The machine that produced this problem was a P4. I checked on an > older machine (3) and it didn't have this problem. Here's an example > where the proving doesn't work: > > C:\ajw\base3>pfgw -f -i -tp -q208496*3^39-1 > PFGW Version 20040608.Win_Dev (Beta 'caveat utilitor') [FFT v23 w/P4] I see the same for this number on the 20040608 version also. The 20040205 version does correctly list this as prime. When I run it with the "broken" PFGW using verbose=true in the pfgw.ini, I get this: Primality testing 208496*3^39-1 [N+1, Brillhart-Lehmer-Selfridge] trial factoring to 65536 Running N+1 test using discriminant 29, base 1+sqrt(29) Adjusting authentication level by 1 for PRIMALITY PROOF non-SSE2 FFT: 176 bit request FFT size=(40,6) - [from (40,23)] Using 'Generic' modular reduction code Running N+1 test using discriminant 29, base 2+sqrt(29) Adjusting authentication level by 1 for PRIMALITY PROOF SSE2 FFT: 176 bit request FFT size=(64,4) - [from (64,23)] Using 'Generic' modular reduction code 208496*3^39-1 is composite (0.0066s+0.0002s) a 64 limb FFT with 4 bit limbs. Something is VERY VERY wrong there. In the working (v 20040205), this is the FFT's chosen Primality testing 208496*3^39-1 [N+1, Brillhart-Lehmer-Selfridge] trial factoring to 65536 Running N+1 test using discriminant 29, base 1+sqrt(29) Using non-SSE2 FFT Adjusting authentication level by 1 for PRIMALITY PROOF Reduced from FFT(40,23) to FFT(40,22) Reduced from FFT(40,22) to FFT(40,21) Reduced from FFT(40,21) to FFT(40,20) Reduced from FFT(40,20) to FFT(40,19) Reduced from FFT(40,19) to FFT(40,18) Reduced from FFT(40,18) to FFT(40,17) Reduced from FFT(40,17) to FFT(40,16) 176 bit request FFT size=(40,16) Running N+1 test using discriminant 29, base 2+sqrt(29) I am not sure what went wrong, but I can easily find out why it chose such an obviously wrong FFT size. Jim. =============================================== jim_fougeron Message 13 of 43 Jun 17, 2004 ----------------------------------------------- There is a new build (dev build) of PFGW contained within the OpenPFGW Yahoo group's file folder. This version fixes the bug listed here, and it also should now again be buildable under Linux. Greg is working on making sure that is so, and building a "static" linked pfgw for Linux. Thanks Andrew for reporting this. Jim. =============================================== Ulrich Thiel Message 14 of 43 Jun 18, 2004 ----------------------------------------------- Hi! The newest release works fine! Great work! The problem with the configure scripts was Gentoo Linux specific. I've found a message describing the solution in the archive: You have to unset the CC and CXX global variables. Perhaps you can add #gentoo specific unset CC unset CXX to the beginning of the main configure script!? After this the script works fine. Ulrich Thiel =============================================== masserto Message 15 of 43 Sep 3, 2004 ----------------------------------------------- Dear Guido, I recently found that 2*k*3^n-1 is always composite if k=739171331147778631 The "covering set" for this Riesel number is: {5,7,13,17,19,37,73,97,577,757,769} Has anyone found a smaller k so that 2*k*3^n-1 is always composite? Best regards, Tom Masser =============================================== Guido Smetrijns Message 16 of 43 Sep 4, 2004 ----------------------------------------------- Dear Tom, My congratulations and applause for your 18-digit Riesel number in base 3, quite an improvement on my initial 27-digit finding! I guess it must have kept you busy for some time! However, I think there is a possibility that your current record might be broken again. After your announcement I had another look at the problem and discovered a covering set with only 10 elements, compared to the 11-element covering set of yours. This set is {5,7,13,19,37,73,109,433,757,8209} and the corresponding product is 6848415653309939345, which is about half of the product of {5,7,13,17,19,37,73,97,577,757,769} = 12933267788230776805. This means that, statistically, the solutions to a system of linear congruences (via CRT) will also be smaller. With this in mind, I "manually" solved as a test-case one such system (10 congruences) and came up with a 19-digit solution k=2595760713702375057. Since the number of possible systems of linear congruences can be quite high, I suppose we still haven't hit the bottom line here In order to work this thing further out, however, I would need two things: First, a way to generate all possible systems of linear congruences for a given covering set, and secondly, a "tool" or "CRT-applet" for easily solving such a system with large integer coefficients. So far I haven't found anything on the internet yet... And maybe still smaller covering sets are just waiting to be discovered... Any help is welcome! Kind regards, Guido Smetrijns Leuven - Belgium. =============================================== Mikael Klasson Message 17 of 43 Sep 4, 2004 ----------------------------------------------- Hi, here's a smaller 14-digit k with 9 elements in the covering set: k=17630689120601, covering set {5,7,13,17,19,37,41,193,757} It's shamelessly based off of Tom's solution, and can probably be optimized further. Cheers, Mikael =============================================== Guido Smetrijns Message 18 of 43 Sep 4, 2004 ----------------------------------------------- Hi Mikael, Congrats to you too! But, as you suspected already yourself, still not optimal : 15618563306548 is the one to beat now (same covering set). Who's next? Let's aim for a 13-digit one! Can the covering set still be improved? Hmmm, I doubt it... Greetz, Guido. =============================================== Mikael Klasson Message 19 of 43 Sep 5, 2004 ----------------------------------------------- Hi Guido, here's an 11-digit k for you! k=31532322469, same covering set. Cheers, Mikael =============================================== andrew_j_walker Message 20 of 43 Sep 5, 2004 ----------------------------------------------- --- In primeform@yahoogroups.com, "masserto" <masserto@y...> wrote: > > Dear Guido, > > I recently found that 2*k*3^n-1 is always composite if > > k=739171331147778631 > > The "covering set" for this Riesel number is: > > {5,7,13,17,19,37,73,97,577,757,769} > > Has anyone found a smaller k so that 2*k*3^n-1 is always composite? > > Best regards, > Tom Masser > Nice work, that's smaller than any I remember seeing. Have you looked at the plus case? Andrew =============================================== andrew_j_walker Message 21 of 43 Sep 5, 2004 ----------------------------------------------- --- In primeform@yahoogroups.com, Mikael Klasson <mikkl605@s...> wrote: > Hi Guido, > > here's an 11-digit k for you! > > k=31532322469, same covering set. > > Cheers, > Mikael > Brilliant! Now we're getting to the range where testing all lower k might be possible. I've been testing plus and minus up to 10 million, however going up to 10^9 would be reasonable, or even higher if the range of n is limited to start with. Andrew =============================================== Payam Samidoost Message 22 of 43 Sep 5, 2004 ----------------------------------------------- > > here's an 11-digit k for you! > > k=31532322469, same covering set. > > Brilliant! Now we're getting to the range where testing all lower k > might be possible. I've been testing plus and minus up to 10 million, > however going up to 10^9 would be reasonable, or even higher > if the range of n is limited to start with. > Do NOT forget the dual approach, as it can sieve out MANY candidates. Payam =============================================== andrew_j_walker Message 23 of 43 Sep 5, 2004 ----------------------------------------------- --- In primeform@yahoogroups.com, "Payam Samidoost" <sami@a...> wrote: > > > > here's an 11-digit k for you! > > > k=31532322469, same covering set. > > > > Brilliant! Now we're getting to the range where testing all lower k > > might be possible. I've been testing plus and minus up to 10 million, > > however going up to 10^9 would be reasonable, or even higher > > if the range of n is limited to start with. > > > > Do NOT forget the dual approach, as it can sieve out MANY candidates. > > Payam Thanks for the reminder about these, if I go much higher I will use this however up to 10 million I'll try to get as many primes as possible in the k*3^n+-1 form. I'll have to modify the script I was using to output the dual primes found as well. Unfortuneately the duals are more limited as you have to use a general prime test on them rather than a proth-like test. Andrew =============================================== Payam Samidoost Message 24 of 43 Sep 5, 2004 ----------------------------------------------- Hi Andrew > Thanks for the reminder about these, if I go much higher I will > use this however up to 10 million I'll try to get as many primes as > possible in the k*3^n+-1 form. I'll have to modify the script I was > using to output the dual primes found as well. Unfortuneately > the duals are more limited as you have to use a general prime test > on them rather than a proth-like test. Small exponent duals (which can easily be *proved* prime) can sieve out MANY of the candidates (up to %80). I strongly recommend testing them just from the begining. Even large exponent duals which remain *probable* primes, still can narraw the search FAR faster than the positive side. I recall that we are doing some practical work here, to find the smallest k. Considering the duals may be a great practical improvement. Yours, Payam =============================================== andrew_j_walker Message 25 of 43 Sep 5, 2004 ----------------------------------------------- --- In primeform@yahoogroups.com, "Payam Samidoost" <sami@a...> wrote: > Hi Andrew > > > Thanks for the reminder about these, if I go much higher I will > > use this however up to 10 million I'll try to get as many primes as > > possible in the k*3^n+-1 form. I'll have to modify the script I was > > using to output the dual primes found as well. Unfortuneately > > the duals are more limited as you have to use a general prime test > > on them rather than a proth-like test. > > Small exponent duals (which can easily be *proved* prime) can sieve out MANY > of the candidates (up to %80). I strongly recommend testing them just from > the begining. > > Even large exponent duals which remain *probable* primes, still can narraw > the search FAR faster than the positive side. > > I recall that we are doing some practical work here, to find the smallest k. > Considering the duals may be a great practical improvement. > I intend to do the duals as well, what I plan is: a) Search both without duals up to k=10^7, what I'm doing now b) Search both with duals up to a higher k. I initially did this a year or so ago up to around 1 million, however using the faster pfgw with scripting it should be possible to go a lot further (maybe 10^9 or more). Andrew =============================================== Mikael Klasson Message 26 of 43 Sep 6, 2004 ----------------------------------------------- Hi, here's an 11-digit candidate for the 2*k*3^n+1 case: k=62525488043, covering set {5,7,13,17,19,37,41,193,757} Does anyone have a smaller k? Mikael =============================================== Guido Smetrijns Message 27 of 43 Sep 6, 2004 ----------------------------------------------- First of all my congratulations and standing ovation for Mikael Klasson !!! Very "klassy", Mikael! I never expected such a low k-value was possible! Meanwhile my poor old PC has been able to proof overnight that k=31532322469 is indeed the minimal value for the covering set {5,7,13,17,19,37,41,193,757}. If this k is still to be improved, it will thus require an even more appropriate covering set !! Secondly, we might indeed start to think about trying the other way, from the bottom up. If I understand well, Andrew Walker has already searched all k values less then 10 million, both in the plus and the minus case. I hereby assume that all k-values < 10,000,000 did not produce a possible Riesel/Sierpinski number, and furthermore I assume that the "tested" expression was k*3^n+-1 in stead of 2k*3^n+-1, so all tested k-values were actually already even to start with. Andrew, can you confirm this?. The reason for pointing this out is the fact that I did some testing myself above 10 million in the minus case. The first possible Riesel candidate I discovered was k=10,789,522. More accurately, the expression 10789522*3^n-1 is composite for all n<620000. It will thus require a "new record base-3 Proth prime" to demolish the current Riesel character of this number, if at all possible. I personally gave up exploring this value any further, but if someone else wants to give it a try, I would be glad to pass him/her my previous results and my sieving files (sieved to a depth of 25*10^9). Kind regards, Guido. =============================================== andrew_j_walker Message 28 of 43 Sep 6, 2004 ----------------------------------------------- --- In primeform@yahoogroups.com, "Guido Smetrijns" <guido@s...> wrote: > First of all my congratulations and standing ovation for Mikael Klasson !!! > Very "klassy", Mikael! > > I never expected such a low k-value was possible! > Meanwhile my poor old PC has been able to proof overnight that k=31532322469 > is indeed the minimal value for the covering set > {5,7,13,17,19,37,41,193,757}. If this k is still to be improved, it will > thus require an even more appropriate covering set !! > > Secondly, we might indeed start to think about trying the other way, from > the bottom up. > If I understand well, Andrew Walker has already searched all k values less > then 10 million, both in the plus and the minus case. Only up to about k=40,000 or so, more for k<10^6 > I hereby assume that all k-values < 10,000,000 did not produce a possible > Riesel/Sierpinski number, and furthermore I assume that the "tested" > expression was k*3^n+-1 in stead of 2k*3^n+-1, so all tested k- values were > actually already even to start with. Andrew, can you confirm this?. I only used even k, up to moderate limits. When I start a new search I'll go deeper than I previously did for the dual case. > The reason for pointing this out is the fact that I did some testing myself > above 10 million in the minus case. The first possible Riesel candidate I > discovered was k=10,789,522. More accurately, the expression 10789522*3^n-1 > is composite for all n<620000. It will thus require a "new record base-3 > Proth prime" to demolish the current Riesel character of this number, if at > all possible. > I personally gave up exploring this value any further, but if someone else > wants to give it a try, I would be glad to pass him/her my previous results > and my sieving files (sieved to a depth of 25*10^9). > > Kind regards, > Guido. > Guido, that matches with the lowest k I found as well, I tested up to about n=200,000. However I didn't go very far with the dual of abs(k-3^n). Does anyone know what to do if this dual turns out to be 1,3,5 or 7 ?? Andrew =============================================== andrew_j_walker Message 29 of 43 Sep 6, 2004 ----------------------------------------------- > Guido, that matches with the lowest k I found as well, I tested up > to about n=200,000. However I didn't go very far with the dual > of abs(k-3^n). Does anyone know what to do if this dual turns out to > be 1,3,5 or 7 ?? > > Andrew Actually a post has just come through to the number theory mailing list from Eric Brier stating: "If a number n is such that n*2^k + 1 is composite for all k due to a covering set of primes, this set of primes ensures also that 2^k + n is composite for all k, maybe with very special case of 2^k + n = one of the covering primes" So how can we be sure in the dual case we don't eliminate one of the special cases by mistake? Andrew =============================================== Mikael Klasson Message 30 of 43 Sep 7, 2004 ----------------------------------------------- Thank you, Guido. For the curious, I've searched through all k less than 31532322469, checking if any of them has a covering set under the following conditions: 1. each prime p in the set is smaller than 10^6 2. each prime p in the set has order(3,p)<=200 3. k != 0 (mod 3) No match was found until 31532322469. In retrospect the order bound of 200 was probably a bit low. Nevertheless, the chance of a smaller k seems very slim to me. Cheers, Mikael =============================================== andrew_j_walker Message 31 of 43 Sep 8, 2004 ----------------------------------------------- Below is the scriptify script based on bits from other people and myself I was going to use to investigate this further. Unfortuneately it seems a bit too slow to use up to large k, can anyone offer suggestions for speeding it up? Note that the n=300 I chose to split between using the isPRP() function and pfgws PRP() test is arbitary. For the vast majority of k a pseudoprime will be found for n<300. From a similar script I was using checking the non-dual case, pfgw is by far faster than something like pari for larger n, but maybe pari is faster for smaller n? At present isPRP() uses only a base 5 test, as the comment suggests I also have tried this using a number of bases, reducing the number of bases had little change on the speed. Using the 5 bases followed by factoring up to 10000 should eliminate all but the stubbornest composites! Andrew int isPRP(int val) { int prp5; int prp101; int prp137; int prp11; int prp67; factorize(val,2,100); if(FACTORFOUND>1) { if (FACTORFOUND==val) { return 1; } return 0; } prp5=powmod(5,val-1,val); /* check for PRP-5 */ if(prp5==1) { factorize(val,2,10000); if(FACTORFOUND>1) { if (FACTORFOUND==val) { return 1; } return 0; } return 1; /* number is PRP-5,PRP-137,PRP-101,PRP-11 and PRP-67 */ } return 0; } file log1000; file logmissing; file logmissing2; log1000 = fopen("duals.txt", "a"); logmissing = fopen("no_dual_plus.txt", "a"); logmissing2 = fopen("no_dual_minus.txt", "a"); int k = 2; int nextoutput = k+1000; int n; int tmp1; int tmp2; int tmp3; int tmp4; int tmp5; int missing=0; int found; for (; k <= 10000000; k = k + 2) { if (k % 3) { found=0; for (n = 1; n < 8192; n = n + 1) { tmp1 = k*3^n+1; tmp2 = k+3^n; if (n<300) { tmp3 = isPRP(tmp1); tmp4 = isPRP(tmp2); } if (n>=300) { tmp3 = PRP(tmp1); tmp4 = PRP(tmp2); } if (tmp3 == 1) { found=1; if (n >= 1000) { fprintf(log1000, "%d*3^%d+1", k, n); } n=1000000; /* to get out of the loop */ } if (tmp4 == 1) { if (found == 0) { found=1; if (n >= 1000) { fprintf(log1000, "%d+3^%d", k, n); } } n=1000000; /* to get out of the loop */ } } if (found == 0) { missing = missing + 1; fprintf(logmissing, "%d", k); } found=0; for (n = 1; n < 8192; n = n + 1) { tmp1 = k*3^n-1; tmp2 = 3^n-k; if (tmp2<0) { tmp2= -tmp2; } if (n<300) { tmp3 = isPRP(tmp1); tmp4 = isPRP(tmp2); } if (n>=300) { tmp3 = PRP(tmp1); tmp4 = PRP(tmp2); } if (tmp3 == 1) { found=1; if (n >= 1000) { fprintf(log1000, "%d*3^%d-1", k, n); } n=1000000; /* to get out of the loop */ } if (tmp4 == 1) { if (found == 0) { found=1; if (n >= 1000) { fprintf(log1000, "abs(%d-3^%d)", k, n); } } n=1000000; /* to get out of the loop */ } } if (found == 0) { missing = missing + 1; fprintf(logmissing2, "%d", k); } if (k > nextoutput) { nextoutput += 1000; printf ("Finished k=%d (%d not found)\n", k, missing); } } } =============================================== jim_fougeron Message 32 of 43 Sep 9, 2004 ----------------------------------------------- The biggest speedup is to not use PFGW, but to use a custom C program with GMP. With C (or C++), you can do true sieving, and thus not speed up by an order of magnitude. I would recommend GMP to test up to like n=300. Any k's which are "missing" from that, can be run through a script like seen below. That script would read from a file, vs a for loop. Very few k's would live past 300. I could probably pump out a program based upon what I see below (and using the GMP dll's in our file folders), in a very short time (an hour or so). Jim. =============================================== jim_fougeron Message 33 of 43 Sep 9, 2004 ----------------------------------------------- I have written a very quick C program, that does this: 1st it builds a bit array of all primes from 3 to 2^31-1. It then loops over k's, looping over n's. It tests k+3^n until that value is over 2^31, or a prime is found. If no prime is found, it then tests k*3^n+1 up to where k*3^n+1 exceeds 2^31 or a prime is found. If no prime found, then it outputs this to a log file. Then it does the same for 3^n-k (abs of course), and k*3^n-1. Stopping if the results exceeds 3^31 or if a prime is found. The program takes about 30 seconds to fill the prime bitmap. It then takes about .5s to compute k's from 2 to 1000000. There were 3200 values (1676 of the +'s and 1524 of -'s) which no small prime was found. Those are all of the values left, and these could be "fed" back into a PFGW script, or an "extension" of my program. NOTE that as k gets larger, then only checking the expressions that are under 2^31 is not adequate, however, it does still cut down a run of 100,000,000 into only 1657756 candidates. I am sure that if I did some real quick testing using GMP (up to say n=50), there would be only a few hundred left. Jim. =============================================== andrew_j_walker Message 34 of 43 Sep 9, 2004 ----------------------------------------------- Thanks Jim, I wouldn't mind getting a copy of your siever to try running on some larger k ranges. Since my last post I remembered that I wrote a simple gmp program last year which just did prp tests up to n=1023 without any trail division (although mpz_probab_prime_p() does internally trial divide to some small limit). I'll put a copy at the bottom if anyone wants to do comparisons. I only ever ran this for the minus case, the output looks like 3606326 5539472 10789522 15719414 16184054 . . . 960068014 962539444 962798786 964544428 (598 entries). Before this I did a search for both up to n=500 and k=100 million. (using pari). I did a lot of work on the k.3^n+-1 values finding primes for nearly all of them, but almost no work on k+3^n or abs (k-3^n). --- In primeform@yahoogroups.com, "jim_fougeron" <jfoug@c...> wrote: > I have written a very quick C program, that does this: > > 1st it builds a bit array of all primes from 3 to 2^31-1. > > It then loops over k's, looping over n's. > > It tests k+3^n until that value is over 2^31, or a prime is found. > If no prime is found, it then tests k*3^n+1 up to where k*3^n+1 > exceeds 2^31 or a prime is found. If no prime found, then it outputs > this to a log file. > > Then it does the same for 3^n-k (abs of course), and k*3^n-1. > Stopping if the results exceeds 3^31 or if a prime is found. > > The program takes about 30 seconds to fill the prime bitmap. It > then takes about .5s to compute k's from 2 to 1000000. There were > 3200 values (1676 of the +'s and 1524 of -'s) which no small prime > was found. Those are all of the values left, and these could > be "fed" back into a PFGW script, or an "extension" of my program. > > NOTE that as k gets larger, then only checking the expressions that > are under 2^31 is not adequate, however, it does still cut down > a run of 100,000,000 into only 1657756 candidates. I am sure that > if I did some real quick testing using GMP (up to say n=50), there > would be only a few hundred left. > > Jim. > #include <stdio.h> #include <stdlib.h> #include <gmp.h> main (int argc, char **argv) { long n,fc,done; mpz_t k,x,y; FILE * NumStream; mpz_init(k); mpz_init(x); mpz_init(y); mpz_init_set_ui(k,2); while(1) { done=0; mpz_set_ui(y,1); for(n=1;n<=1023;n++) { mpz_mul_ui(y,y,3); /* y=3^n */ mpz_mul(x,y,k); mpz_sub_ui(x,x,1); /* x=k*3^n-1 */ if( (mpz_probab_prime_p(x,10)>0) ) { done=1; n=1023; } if(done==0) { mpz_sub(x,k,y); /* x=k-y=k-3^n */ mpz_abs(x,x); /* x=abs(k-3^n) */ if( (mpz_probab_prime_p(x,10)>0) ) { done=1; n=1023; } } } if(done==0) { gmp_printf("%Zd\n",k); NumStream = fopen("dualminus.txt","a"); gmp_fprintf(NumStream,"%Zd\n",k); fclose(NumStream); } mpz_add_ui(k,k,2); if( mpz_divisible_ui_p(k,3) ) mpz_add_ui(k,k,2); } mpz_clear(k); mpz_clear(x); mpz_clear(y); return(0); } =============================================== jim_fougeron Message 35 of 43 Sep 9, 2004 ----------------------------------------------- Here are all of the 3^n + and 3^n - that are left over. They are "clean" up to n=10000. All k's up to 100 million were tested. The pfgw input file will stop processing a k, when a PRP is found for it (Of course you will need to make sure the prp is prime later). It is setup to test from n=10000 to n=20000. After you get to 20k, then remove any "k" values found (the a: param), adjust the n value to a higher range (the b: value), and re-run. If you want, I can push from the k range 10^8 up to 10^9 pretty easily. I will not test "as" deeply (probably cutoff at n=2000, but there should not be more than 100 at that level. ABC2 $a+3^$b | $a*3^$b+1 // {number_primes,$a,1} a: in { 2621746 11013584 48886226 59054564} b: from 10000 to 20000 ABC2 3^$b-$a | $a*3^$b-1 // {number_primes,$a,1} a: in { 10789522 23375192 29140796 36730222} b: from 5000 to 10000 NOTE there are 4 candidates in the + side, and 4 in the - side So far no k under 10^8 (which I have tested) appear to have a covering set of factors, but the above 8 candidates are VERY sparse. Jim. PS: One nice thing out of this "test" was that your script found a bug in the new dev version of PFGW (which was not in the "older" stable version). In the new version, I have de-coupled the expression parser from the rest of the program (am still working on that), and have created an "easy_eval" set of functions. Well there was some behavior missing (calling the easy_eval_GetInteger() with a literal number, and not a "symbol" number was failing). =============================================== andrew_j_walker Message 36 of 43 Sep 9, 2004 ----------------------------------------------- Below the numbers I've put some data on which of these k values I found primes for. There could of course be a smaller n prime for the dual form. --- In primeform@yahoogroups.com, "jim_fougeron" <jfoug@c...> wrote: > Here are all of the 3^n + and 3^n - that are left over. They are > "clean" up to n=10000. All k's up to 100 million were tested. > The pfgw input file will stop processing a k, when a PRP is found > for it (Of course you will need to make sure the prp is prime later). > It is setup to test from n=10000 to n=20000. After you get to 20k, > then remove any "k" values found (the a: param), adjust the n value > to a higher range (the b: value), and re-run. > > If you want, I can push from the k range 10^8 up to 10^9 pretty > easily. I will not test "as" deeply (probably cutoff at n=2000, > but there should not be more than 100 at that level. > > ABC2 $a+3^$b | $a*3^$b+1 // {number_primes,$a,1} > a: in { 2621746 11013584 48886226 59054564} > b: from 10000 to 20000 2621746*3^119335+1 is prime (dual tested to 10000) 11013584*3^100378+1 is prime(dual tested to 10000) 48886226*3^51212+1 is prime (dual tested to 10000) 59054564*3^64030 is prime (dual tested to 10000 > ABC2 3^$b-$a | $a*3^$b-1 // {number_primes,$a,1} > a: in { 10789522 23375192 29140796 36730222} > b: from 5000 to 10000 > 10789522*3^n-1 no prime up to n=200000 23375192*3^n-1 no prime up to n=150062 29140796*3^n-1 no prime up to n=150062 36730222*3^15624-1 is prime > NOTE there are 4 candidates in the + side, and 4 in the - side > > So far no k under 10^8 (which I have tested) appear to have a > covering set of factors, but the above 8 candidates are VERY sparse. > > Jim. > > PS: > One nice thing out of this "test" was that your script found a bug > in the new dev version of PFGW (which was not in the "older" stable > version). In the new version, I have de-coupled the expression > parser from the rest of the program (am still working on that), and > have created an "easy_eval" set of functions. Well there was some > behavior missing (calling the easy_eval_GetInteger() with a literal > number, and not a "symbol" number was failing). > > =============================================== andrew_j_walker Message 37 of 43 Sep 9, 2004 ----------------------------------------------- --- In primeform@yahoogroups.com, "andrew_j_walker" <ajw01@u...> > > 10789522*3^n-1 no prime up to n=200000 > 23375192*3^n-1 no prime up to n=150062 below was wrong before. 29140796*3^n-1 no prime up to n=200000 > 36730222*3^15624-1 is prime > > =============================================== Robert Message 38 of 43 Oct 1, 2006 ----------------------------------------------- --- In primeform@yahoogroups.com, "andrew_j_walker" <ajw01@...> wrote: > > --- In primeform@yahoogroups.com, "masserto" <masserto@y...> wrote: > > > > Dear Guido, > > > > I recently found that 2*k*3^n-1 is always composite if > > > > k=739171331147778631 > > > > The "covering set" for this Riesel number is: > > > > {5,7,13,17,19,37,73,97,577,757,769} > > > > Has anyone found a smaller k so that 2*k*3^n-1 is always composite? > > > > Best regards, > > Tom Masser > > > > Nice work, that's smaller than any I remember seeing. Have you looked > at the plus case? > > Andrew There has been a delay since this was posted. Regarding the Sierpinski case: One such covering set is [13,5,7,41,73,17,193,6481,97,577] which have multiplicative order base 3 of 3,4,6,8,12,16,16,24,48,48, all of which are factors of 48. Using CRM provides the following k which provides composite 2*k*3^n+1 for all n: 36785490291994693 I am by no means convinced this is the smallest k but it might be as it is 20 times smaller than the lowest known Riesel. It will be a very hard problem to prove this is the lowest 2*k never prime. The corresponding Riesel associated with this covering set provides a larger value of k than the Tom's Riesel value. A nice series for OEIS would be 78557,36785490291994693,66741,159986.... Mooted Sierpinski numbers base a=2,3,4,5..., where to be the k value the Sierpinski must be multiplied by all primes which have multiplicative order base a of 1 (to elimiate trivial results). Anyone up to extend this series as a challenge? Regards Robert Smith =============================================== David Broadhurst Message 39 of 43 Oct 1, 2006 ----------------------------------------------- --- In primeform@yahoogroups.com, "Robert" <rw.smith@...> wrote: > A nice series for OEIS would be > 78557,36785490291994693,66741,159986.... > Mooted Sierpinski numbers base a=2,3,4,5... Please send it to Neil, with "mooted" ==> conjectured. Nice work with a=3, thanks, Robert. David =============================================== Robert Message 40 of 43 Oct 2, 2006 ----------------------------------------------- --- In primeform@yahoogroups.com, "David Broadhurst" <d.broadhurst@...> wrote: > > --- In primeform@yahoogroups.com, "Robert" <rw.smith@> wrote: > > > A nice series for OEIS would be > > 78557,36785490291994693,66741,159986.... > > Mooted Sierpinski numbers base a=2,3,4,5... > > Please send it to Neil, with "mooted" ==> conjectured. Done and dusted, although I am no expert on finding suitable references Regards Robert Smith =============================================== David Broadhurst Message 41 of 43 Oct 2, 2006 ----------------------------------------------- --- In primeform@yahoogroups.com, "Robert" <rw.smith@...> wrote: > > 78557,36785490291994693,66741,159986.... > Done and dusted Did you remember to multiply 36785490291994693 by 2 ? David =============================================== Robert Message 42 of 43 Oct 2, 2006 ----------------------------------------------- --- In primeform@yahoogroups.com, "David Broadhurst" <d.broadhurst@...> wrote: > > --- In primeform@yahoogroups.com, "Robert" <rw.smith@> wrote: > > > > 78557,36785490291994693,66741,159986.... > > Done and dusted > > Did you remember to multiply 36785490291994693 by 2 ? > > David > Defined it in a different way, so that each of the numbers in the series must be multiplied by all primes with multiplicative order base b of 1 to get to a k. Regards Robert =============================================== Robert Message 43 of 43 Jan 7, 2007 ----------------------------------------------- --- In primeform@yahoogroups.com, "Robert" <rw.smith@...> wrote: > > > Regarding the Sierpinski case: > > One such covering set is [13,5,7,41,73,17,193,6481,97,577] which have > multiplicative order base 3 of 3,4,6,8,12,16,16,24,48,48, all of which > are factors of 48. Using CRM provides the following k which provides > composite 2*k*3^n+1 for all n: > > 36785490291994693 > > I am by no means convinced this is the smallest k but it might be as > it is 20 times smaller than the lowest known Riesel. > Aiaiai, doing further research on this shows that, in the primenumbers group the following got posted, 20 times smaller than my value: The minimal k with covering set of size 48 is: k=3574321403229074 The modulus M is 2*5*7*13*17*41*73*97*193*769*6481 Source: Jack Brennen May 17, 2002, Yahoo Primenumbers Sorry about that, but it is good to know that the problem is 20 times easier to solve. Regards Robert Smith =============================================== Cached by Georg Fischer at Nov 14 2019 12:47 with clean_yahoo.pl V1.4