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2022-10-18, 12:45
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#23 |
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"ม้าไฟ"
May 2018
21616 Posts |
Below are 2 cyclic LCS sets related to post #18 at https://mersenneforum.org/showpost.p...2&postcount=18.
The first cyclic LSC set contains 467 distinct cyclic LCSs for all 51 exponents of known Mersenne primes. Code:
1 10 11 010 011 101 110 111 0010 0011 0110 0111 1000 1010 1011 1100 1101 1110 1111 00010 00011 00101 00110 00111 01000 01001 01010 01011 01100 01101 01110 01111 10000 10001 10010 10011 10100 10110 10111 11001 11010 11011 11100 11101 11110 11111 000010 000011 000110 001001 001011 001100 001101 001110 001111 010001 010011 010100 010110 010111 011000 011001 011010 011011 011100 011101 011110 011111 100011 100101 100110 100111 101000 101001 101011 101100 101101 101110 101111 110001 110010 110011 110100 110101 110110 110111 111000 111001 111010 111011 111100 111101 111110 111111 0000010 0000011 0000111 0001011 0001100 0001101 0010010 0010100 0011000 0011001 0011010 0011011 0011100 0011101 0011110 0011111 0100000 0100011 0100101 0100110 0100111 0101000 0101011 0101100 0101101 0101111 0110000 0110001 0110011 0110101 0110110 0110111 0111000 0111001 0111010 0111011 0111100 0111101 0111111 1000001 1000011 1000110 1000111 1001011 1001100 1001101 1001110 1001111 1010000 1010011 1010100 1010110 1011000 1011011 1011100 1011101 1011110 1011111 1100001 1100010 1100110 1100111 1101001 1101010 1101011 1101100 1101101 1101110 1101111 1110001 1110010 1110011 1110100 1110101 1110110 1110111 1111000 1111010 1111011 1111100 1111111 00000011 00001100 00001101 00010011 00011010 00011011 00011100 00011110 00100110 00110000 00110001 00110011 00110100 00110110 00110111 00111001 00111011 00111100 00111101 00111111 01000111 01001000 01001100 01001101 01001110 01010000 01010011 01010101 01011000 01011101 01011111 01100000 01100001 01100010 01100011 01100110 01100111 01101001 01101011 01101100 01101110 01110001 01110011 01110100 01111000 01111010 01111011 01111111 10000000 10000110 10001101 10001110 10010010 10011000 10011001 10011010 10011011 10011101 10011110 10011111 10100000 10100011 10100110 10100111 10101011 10101111 10110000 10110001 10110011 10110101 10111101 10111110 11000011 11000101 11001001 11001101 11001111 11010001 11010010 11010011 11010100 11010101 11011000 11011100 11011101 11100010 11100011 11100110 11100111 11101001 11101010 11101100 11101101 11101110 11110001 11110011 11110100 11110101 11110110 11110111 000001111 000010011 000100110 000110011 001001011 001001101 001011111 001100000 001100011 001100110 001101001 001101011 001101100 001101101 010000011 010001110 010011000 010011011 010011101 010101011 010110001 010111110 010111111 011000011 011000101 011001001 011001100 011001101 011011001 011011110 011100011 011101000 011101100 011110100 011111100 011111110 100000001 100011000 100011101 100101110 100110001 100110010 100110100 100110110 100110111 100111011 101000000 101000111 101001100 101011111 101100010 101100110 101101010 101110011 101110110 101111010 101111101 110000110 110001001 110001010 110010010 110010111 110011001 110011010 110011011 110100000 110100100 110100111 110101011 110101111 110110101 110110111 110111000 111000110 111001101 111001111 111010010 111010011 111011000 111011100 111100010 111101001 111101101 111110001 111110101 111111001 111111111 0011011110 0100011101 0100111011 0101000011 0101011001 0110011011 0110100111 0110101101 0110101111 0110110101 0110111101 0111011000 0111100110 1000110000 1000110001 1000111010 1001001011 1001100010 1001110011 1001111011 1010011000 1010011101 1010100110 1011001101 1011011101 1011100110 1011101000 1011101100 1100001100 1100010011 1100011101 1100110110 1100111011 1101001111 1101011110 1101100010 1101110001 1101110110 1110000110 1110001001 1110011110 1110100011 1111010011 1111111111 00000001111 00101111110 00110010111 00110011110 00110111000 00111011000 01101101011 01110100011 01110110110 01111010011 10011101110 10011110110 10100110001 10110001010 11000101000 11000110110 11000111010 11001011111 11001101000 11001111011 11011010101 11100110011 11101000110 11110100111 000100111011 001001011000 010010001110 010011000001 010011101110 011000111010 011101100010 100000110111 100011101001 100110110101 100111011100 101100110110 101110011010 110101001100 110110001010 111010011000 0000011110100 0000110001100 0001100011001 0101011001101 0110001001110 1100011000110 1100110110101 1110110001010 1110111000110 1111111001111 10011011010101 10011101100001 11000111010100 11010011101110 11011010101010 100001101001000 110101111101001 110110001010000 0100111011100110 1000011100011101 1011000101011001 Code:
(* Wolfram code *)
MpData = {2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279, 2203, 2281, 3217, 4253, 4423, 9689, 9941, 11213, 19937, 21701, 23209, 44497, 86243, 110503, 132049, 216091, 756839, 859433, 1257787, 1398269, 2976221, 3021377, 6972593, 13466917, 20996011, 24036583, 25964951, 30402457, 32582657, 37156667, 42643801, 43112609, 57885161, 74207281, 77232917, 82589933};
nMp = Length[MpData]; LCSa = ConstantArray[0, nMp*(nMp - 1)/2]; base = 2; count = 0;
ic = 0; While[ic < nMp, ic++; intlen1 = Length[IntegerDigits[MpData[[ic]], base]]; s1 = IntegerDigits[MpData[[ic]], base, intlen1]; s1 = Join[s1, s1];
jc = 0; While[jc < nMp, jc++; If[ic < jc, intlen2 = Length[IntegerDigits[MpData[[jc]], base]]; s2 = IntegerDigits[MpData[[jc]], base, intlen2]; s2 = Join[s2, s2];
cs = LongestCommonSubsequence[s1, s2];
If[cs != {}, fd = cs; mc = 1; kc = 0; While[kc < count, kc++; If[LCSa[[kc]] == fd, mc = 0;];]; If[mc == 1, count++; LCSa[[count]] = fd;];];
];];];
lcs = ConstantArray[0, count]; kc = 0; While[kc < count, kc++; lcs[[kc]] = LCSa[[kc]];]; lcs = Sort[lcs];
kc = 0; While[kc < count, kc++; Print[IntegerString[lcs[[kc]], base]];]; Print[count];
Code:
1 10 11 010 011 101 110 111 0011 0101 0110 0111 1000 1010 1011 1100 1101 1110 1111 00010 00100 00101 00110 00111 01000 01001 01010 01011 01100 01101 01110 01111 10001 10010 10011 10100 10110 10111 11000 11001 11010 11011 11100 11101 11110 11111 000010 000011 000110 000111 001001 001010 001011 001100 001101 001110 001111 010011 010101 010110 010111 011000 011001 011010 011011 011100 011101 011110 011111 100000 100010 100011 100110 100111 101000 101001 101011 101100 101101 101110 101111 110000 110001 110010 110011 110101 110110 110111 111000 111001 111010 111011 111100 111101 111110 111111 0000010 0000101 0001001 0001100 0001101 0010010 0010011 0010110 0010111 0011000 0011001 0011011 0011100 0011101 0011110 0011111 0100000 0100011 0100100 0100110 0101000 0101001 0101011 0101100 0101101 0101110 0101111 0110001 0110011 0110101 0110110 0110111 0111001 0111011 0111101 0111111 1000110 1000111 1001011 1001100 1001101 1001110 1010111 1011000 1011010 1011101 1011110 1011111 1100000 1100001 1100010 1100100 1100101 1100110 1100111 1101001 1101010 1101011 1101100 1101101 1101110 1101111 1110001 1110010 1110011 1110100 1110101 1110110 1110111 1111001 1111010 1111011 1111101 1111111 00010110 00011001 00011010 00011011 00011101 00011110 00100110 00101100 00101111 00110001 00110011 00110110 00110111 00111010 00111011 00111100 00111111 01000110 01000111 01001100 01001101 01010110 01010111 01011000 01011001 01011110 01011111 01100000 01100001 01100011 01100110 01100111 01101001 01101010 01101100 01101101 01101111 01110001 01110010 01110011 01110110 01111000 01111001 01111101 01111111 10000000 10001100 10001101 10001110 10001111 10010010 10010111 10011000 10011001 10011010 10011100 10011101 10011111 10100011 10100110 10101111 10110001 10110011 10110101 10111011 10111101 10111111 11000001 11000010 11000011 11000101 11000111 11001001 11001011 11001100 11001101 11001111 11010010 11010011 11010100 11010101 11010110 11011000 11011001 11011100 11011101 11011110 11011111 11100001 11100010 11100110 11100111 11101000 11101001 11101010 11101100 11101101 11101111 11110000 11110011 11110100 11110101 11111000 11111001 11111010 000000011 000000101 000001011 000001101 000011001 000101000 000110001 000110111 001001011 001001100 001001101 001011101 001100000 001100011 001100101 001101100 001101110 010001100 010001101 010001111 010011000 010100011 010110011 010111100 010111101 011000000 011000001 011001101 011010100 011011001 011100011 011100100 011101010 011101100 011111100 011111110 100001101 100001110 100011000 100011010 100011011 100011100 100011101 100101100 100101110 100110000 100110001 100110101 100111001 100111011 101000110 101001100 101010110 101011111 101100011 101100110 101101011 101101110 101110001 101110011 101110110 101111001 101111101 110000010 110000110 110001001 110001011 110001110 110010111 110011001 110100100 110101000 110101011 110101111 110110001 110110011 110110101 110111100 110111101 111000101 111001011 111011100 111100111 111101000 111101011 111101101 111110101 111111001 111111111 0000110111 0001100110 0001101110 0010110101 0011011100 0011100100 0011101001 0101111110 0110001011 0110011001 0110011011 0110101010 0110111100 0111100110 0111101011 1000110000 1000110001 1000110011 1000110110 1001001011 1001011111 1001100101 1001101101 1001101110 1001110111 1010001101 1010110011 1010110111 1011000010 1011001101 1011100010 1011110000 1011110011 1100001100 1100011101 1100101111 1100110101 1100110110 1101100110 1101101011 1101101110 1101110001 1110000110 1110011110 1110110000 1111001011 1111111111 00010100011 00011011110 00101110110 00110001011 00110001100 00110010111 00110011110 00110111100 00111010010 01001000111 01011001101 01011111010 01100011101 01101011011 01101110001 01110001001 01110001110 01110111001 01111011011 01111101001 10011101110 10100110001 10111000110 10111001101 10111100110 11001101101 11010101010 11011100011 11011100110 11100101111 11100110011 000110001100 000110100100 000110111000 001001011000 001100011101 001110110000 011000001001 011100001101 011101001100 011111010011 100011000000 101000110101 101010111111 101100011101 110111001101 110111011011 111001101101 111011001100 0011101110011 1001110111001 1100111011100 1111001111111 1111010011000 00001101001000 11000110001100 11101001100011 11111001011111 001101110001001 11100010010110000 Code:
(* Wolfram code *)
MpData = {2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279, 2203, 2281, 3217, 4253, 4423, 9689, 9941, 11213, 19937, 21701, 23209, 44497, 86243, 110503, 132049, 216091, 756839, 859433, 1257787, 1398269, 2976221, 3021377, 6972593, 13466917, 20996011, 24036583, 25964951, 30402457, 32582657, 37156667, 42643801, 43112609, 57885161, 74207281, 77232917, 82589933};
nMp = Length[MpData]; LCSa = ConstantArray[0, nMp*(nMp - 1)/2]; base = 2; count = 0;
ic = 0; While[ic < nMp, ic++; intlen1 = Length[IntegerDigits[MpData[[ic]], base]]; s1 = IntegerDigits[MpData[[ic]], base, intlen1]; s1 = Join[s1, s1];
jc = 0; While[jc < nMp, jc++; If[ic < jc, intlen2 = Length[IntegerDigits[MpData[[jc]], base]]; s2 = IntegerDigits[IntegerReverse[MpData[[jc]],base], base, intlen2]; s2 = Join[s2, s2];
cs = LongestCommonSubsequence[s1, s2];
If[cs != {}, fd = cs; mc = 1; kc = 0; While[kc < count, kc++; If[LCSa[[kc]] == fd, mc = 0;];]; If[mc == 1, count++; LCSa[[count]] = fd;];];
];];];
lcs = ConstantArray[0, count]; kc = 0; While[kc < count, kc++; lcs[[kc]] = LCSa[[kc]];]; lcs = Sort[lcs];
kc = 0; While[kc < count, kc++; Print[IntegerString[lcs[[kc]], base]];]; Print[count];
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2022-10-18, 17:54
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#24 |
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"ม้าไฟ"
May 2018
53410 Posts |
From a binary perspective, there are four thresholds at:
- 27 bits, 1111111111111111111111111112 = 134217727 > M134217689 (factored); - 28 bits, 11111111111111111111111111112 = 268435455 > M268435399 (factored); - 29 bits, 111111111111111111111111111112 = 536870911 > M536870909 (factored); and - 30 bits, 1111111111111111111111111111112 = 1073741823 > M1073741789, https://www.mersenne.ca/exponent/1073741789 (factored). By selecting a number of bits b and a number of LCSs c for concatenation, the product of (the number of combinations of c LCSs in an LCS set) × (the number of permutations of c LCSs) gives how many LCS concatenations could be tested for having b bits and giving a prime number for an exponent of a Mersenne number which remains unfactored/untested in the GIMPS database. This, of course, could be attempted only IF (and this is a big IF) it is assumed that no new LCS (outside the current LCS set) would be needed for the construction of the exponent of the next Mersenne prime (if any). Last fiddled with by Dobri on 2022-10-18 at 18:06 |
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2022-10-19, 21:44
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#25 |
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"ม้าไฟ"
May 2018
10268 Posts |
Here is an illustration concerned with the previous post #24 at https://mersenneforum.org/showpost.p...2&postcount=24.
Let b = 27 and c = 3. The LCS set is the one obtained in post #21 at https://mersenneforum.org/showpost.p...5&postcount=21. The LCS set generates only 27483 27-bit prime exponents in the interval (226 = 67108864, 227 = 134217728). Note that the total number of 27-bit exponents is 3645744. Among the generated exponents are all three known 27-bit exponents of Mersenne primes M74207281, M77232917, and M82589933. The 27483 exponents are generated from the LCS set and saved in a DAT file (see the attached ZIP file) with the Wolfram code shown below. Code:
SetDirectory[NotebookDirectory[]]; fname = NotebookDirectory[] <> "LCSb27c3.dat";
MpData = {2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279, 2203, 2281, 3217, 4253, 4423, 9689, 9941, 11213, 19937, 21701, 23209, 44497, 86243, 110503, 132049, 216091, 756839, 859433, 1257787, 1398269, 2976221, 3021377, 6972593, 13466917, 20996011, 24036583, 25964951, 30402457, 32582657, 37156667, 42643801, 43112609, 57885161, 74207281, 77232917, 82589933};
nMp = Length[MpData]; LCSa = ConstantArray[0, nMp*(nMp - 1)/2]; base = 2; count = 0;
ic = 0; While[ic < nMp, ic++; intlen1 = Length[IntegerDigits[MpData[[ic]], base]]; s1 = IntegerDigits[MpData[[ic]], base, intlen1];
jc = 0; While[jc < nMp, jc++; If[ic < jc, intlen2 = Length[IntegerDigits[MpData[[jc]], base]]; s2 = IntegerDigits[MpData[[jc]], base, intlen2];
cs = LongestCommonSubsequence[s1, s2];
If[cs != {}, fd = cs; mc = 1; kc = 0; While[kc < count, kc++; If[LCSa[[kc]] == fd, mc = 0;];]; If[mc == 1, count++; LCSa[[count]] = fd;];];
];];];
lcs = ConstantArray[0, count]; kc = 0; While[kc < count, kc++; lcs[[kc]] = LCSa[[kc]];]; lcs = Sort[lcs];
base = 2; MnLength = 27; countp = 0; Mna = ConstantArray[0, count^3];
ic = 0; While[ic < count, ic++; s1 = lcs[[ic]];
jc = 0; While[jc < count, jc++; s2 = lcs[[jc]];
kc = 0; While[kc < count, kc++; s3 = lcs[[kc]];
Mns = Join[s1, s2, s3]; Mnl = Length[Mns];
If[(Mnl == MnLength) && (Mns[[1]] == 1) && (Mns[[Mnl]] == 1),
Mni = FromDigits[Mns, base];
If[PrimeQ[Mni] == True, countp++; Mna[[countp]] = Mni;];
];];];];
Mna2 = ConstantArray[0, countp]; kc = 0; While[kc < countp, kc++; Mna2[[kc]] = Mna[[kc]]]; Mna2 = Sort[Mna2];
countp2 = 0; kc = 0; lc = 0; While[(kc < countp) && (lc < countp), kc++; lc++; countp2++; Mna2[[kc]] = Mna2[[lc]]; While[(lc < countp) && (Mna2[[lc]] == Mna2[[lc + 1]]), lc++;];];
Mna3 = ConstantArray[0, countp2]; kc = 0; While[kc < countp2, kc++; Mna3[[kc]] = Mna2[[kc]]];
Print[countp2];
Export[fname, Mna3]
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2022-10-19, 21:59
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#26 | |
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If I May
"Chris Halsall"
Sep 2002
Barbados
261568 Posts |
Quote:
Presume I'm dumber than bricks... What are you trying to tell me? Sincere question. |
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2022-10-19, 22:06
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#27 | |
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"ม้าไฟ"
May 2018
2·3·89 Posts |
Quote:
Last fiddled with by Dobri on 2022-10-19 at 22:08 |
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2022-10-19, 22:22
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#28 | |
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If I May
"Chris Halsall"
Sep 2002
Barbados
2×112×47 Posts |
Quote:
Most of us here are rather serious people. Scientific Method and all of that... |
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2022-10-19, 22:41
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#29 | |
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"ม้าไฟ"
May 2018
21616 Posts |
Quote:
It is a combination of the art of statistics and graph theory. |
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2022-10-19, 22:42
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#30 | |
"TF79LL86GIMPS96gpu17"
Mar 2017
US midwest
32×11×79 Posts |
Quote:
After filtering for both of those considerations, how many of the 27483 candidates remain as not yet shown composite or prime, and which ones? (GIMPS has not yet found a new Mersenne prime as a result of double-checking.) Is there any sound basis to expect common subsequences to reoccur at greater frequency than random chance would provide, or is this in effect a lengthy and ornate exercise of what RDS might call numerology? A way to test that might be the following: remove one of the known Mp in the 27 bit range from the process of re-generating the LCS and candidate sets. Is the removed Mp still in the resulting candidate set? Can it pass that test if a different Mp in the range is omitted from the process instead? How about the third? If it can be demonstrated to pass all three such tests, a case might be made for a little higher priority on factoring and testing the survivors. Last fiddled with by kriesel on 2022-10-19 at 23:16 |
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2022-10-19, 22:46
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#31 | |
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If I May
"Chris Halsall"
Sep 2002
Barbados
2·112·47 Posts |
Quote:
I find it useful to review the submissions of many. It keeps me on my toes and ensures I am still relevant. To be frank... I would not accept you as a student. And would fire you if you were an employee. There's just no "there there". |
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2022-10-19, 22:49
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#32 | |
Apr 2020
3×353 Posts |
Quote:
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2022-10-19, 22:54
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#33 | |
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"ม้าไฟ"
May 2018
10268 Posts |
Quote:
However, my concern is not to congest the GIMPS server, so I may run the code with delays between the server requests and it would take some time before collecting the data. |
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