mersenneforum.org Exponents of Known Mersenne Primes (Numerical Observations)
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 2022-11-22, 06:34 #78 Dobri   "ม้าไฟ" May 2018 22·5·23 Posts The curves of the sorted arrays of ln(|x-y|) and ln(|x|+|y|) of the x-y coordinates of the prime exponents of known Mersenne primes on a modified Ulam clockwise square spiral of 𝜋(p) are shown in the attached image. Note that the two curves of sorted arrays merge at the last several points. Code: (* Wolfram code *) SetDirectory[NotebookDirectory[]]; fname = NotebookDirectory[] <> "MersenneUlamPiXYSortLogAbs.jpg"; MpSx = {1, 1, 0, -1, -1, 0, 1, 2, -2, 2, 3, 2, -3, -4, -7, -9, -9, 11, -12, -10, -11, 18, 7, 24, 25, 16, -34, 34, -51, 56, 41, -123, 131, 156, 164, -144, 234, 345, 115, 577, -563, -2, 359, 357, 425, 532, -296, -150, -248, 1063, 799}; MpSy = {0, -1, -1, -1, 1, 1, 1, -1, 0, 2, -1, -3, -5, 5, 4, -5, 6, -4, -5, 12, 17, 16, 18, -22, -10, 25, -34, -46, 34, 45, 69, 106, -50, -141, 101, -232, 83, 157, 468, 259, 614, -637, -686, 708, 753, -804, -808, -928, -1043, -715, -1097}; nMp = Length[MpSx]; MpSxy = ConstantArray[0, nMp]; MpSxySum = ConstantArray[0, nMp]; ic = 0; While[ic < nMp, ic++; MpSxy[[ic]] = Log[base, Abs[MpSx[[ic]] - MpSy[[ic]]]]; MpSxySum[[ic]] = Log[base, Abs[MpSx[[ic]]] + Abs[MpSy[[ic]]]];]; MpSxy = NumericalSort[MpSxy]; MpSxySum = NumericalSort[MpSxySum]; Show[ListLinePlot[{MpSxy, MpSxySum}, PlotRange -> All, Frame -> True, PlotLabel -> "A graph based on Ulam Clockwise Square Spiral of" PrimePi[p], PlotLegends -> {"ln(|x-y|) (sorted)", "ln(|x|+|y|) (sorted)"}], Frame -> True] Export[fname, Show[ListLinePlot[{MpSxy, MpSxySum}, PlotRange -> All, Frame -> True, PlotLabel -> "A graph based on Ulam Clockwise Square Spiral of" PrimePi[p], PlotLegends -> {"ln(|x-y|) (sorted)", "ln(|x|+|y|) (sorted)"}], Frame -> True]] Attached Thumbnails   Last fiddled with by Dobri on 2022-11-22 at 06:58
2022-11-24, 12:33   #79
kriesel

"TF79LL86GIMPS96gpu17"
Mar 2017
US midwest

53·59 Posts

A small update
Quote:
 Originally Posted by Dobri The prime with coordinates closest to x ≈ 5973 and y ≈ -2658 is M142691651 (Untested)(Factored) with x = 5973 and y = -2654.
Will you be updating previous posts occasionally, as factoring, primality testing, and verification progress?

Last fiddled with by kriesel on 2022-11-24 at 12:34

2022-11-25, 01:06   #80
Dobri

"ม้าไฟ"
May 2018

46010 Posts

Quote:
 Originally Posted by kriesel A small update Will you be updating previous posts occasionally, as factoring, primality testing, and verification progress?
I will post updates occasionally indeed but in new posts without editing old ones, like in the example below.
Quote:
 Originally Posted by Dobri ... M142691651 (now factored) ...

 2022-11-25, 01:28 #81 Dobri   "ม้าไฟ" May 2018 22·5·23 Posts Here are the shortest binary strings that cannot be found as sub-strings in any of the 51 exponents of known Mersenne primes. Code: Seven-bit strings: 0001000 1110010 Eight-bit strings: 00001000 00001010 00010000 00010001 00010110 00011111 00100001 00100010 00101101 00111110 01000010 01000100 01000110 01010010 01011100 01110010 01110101 01111100 10000101 10001000 10001011 10001111 10010101 10101110 10110010 10110011 10110110 11000010 11001000 11001010 11011111 11100000 11100100 11100101 11101011 11101111 11110010 11111011
 2022-11-28, 07:51 #82 Dobri   "ม้าไฟ" May 2018 1CC16 Posts Here are all 216 9-bit binary strings that cannot be found as sub-strings in any of the prime exponents of known Mersenne primes. Code: Nine-bit strings: 000000100 000001000 000001010 000001110 000010000 000010001 000010010 000010100 000010101 000010110 000011001 000011101 000011111 000100000 000100001 000100010 000100011 000100100 000101001 000101100 000101101 000101110 000110101 000110111 000111001 000111100 000111110 000111111 001000000 001000010 001000011 001000100 001000101 001000110 001001001 001001010 001001100 001001111 001010010 001010011 001010100 001010111 001011001 001011010 001011011 001011100 001011110 001101010 001101110 001110000 001110010 001110101 001111000 001111100 001111101 001111110 010000010 010000100 010000101 010000110 010001000 010001001 010001011 010001100 010001101 010001111 010010011 010010101 010010111 010011111 010100010 010100100 010100101 010100111 010101000 010101101 010101110 010110010 010110011 010110101 010110110 010111000 010111001 010111010 010111101 011000010 011000111 011001000 011001010 011001101 011001110 011001111 011010001 011010100 011011001 011011100 011011111 011100000 011100010 011100100 011100101 011101010 011101011 011101111 011110001 011110010 011110101 011111000 011111001 011111011 011111100 100000010 100000101 100001000 100001010 100001011 100001111 100010000 100010001 100010110 100010111 100011010 100011100 100011110 100011111 100100001 100100010 100100111 100101000 100101010 100101011 100101101 100111000 100111010 100111110 101000010 101000100 101000110 101001011 101001101 101010001 101010010 101011100 101011101 101100011 101100100 101100101 101100110 101100111 101101000 101101100 101101101 101110010 101110101 101110111 101111010 101111100 110000001 110000010 110000100 110000101 110001000 110001011 110001101 110001111 110010000 110010001 110010011 110010100 110010101 110010110 110011011 110011100 110100001 110100010 110101000 110101100 110101110 110110010 110110011 110110100 110110110 110111000 110111110 110111111 111000000 111000001 111000010 111000101 111001000 111001001 111001010 111001011 111010000 111010100 111010110 111010111 111011010 111011011 111011110 111011111 111100000 111100011 111100100 111100101 111101011 111101101 111101111 111110000 111110010 111110110 111110111 111111000 111111011 111111110 111111111 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]; base = 2; intlen1 = 9; pcount = 0; ic = -1; While[ic < 2^intlen1, ic++; count = 0; s1 = IntegerDigits[ic, base, intlen1]; jc = 0; While[jc < nMp, jc++; intlen2 = Length[IntegerDigits[MpData[[jc]], base]]; s2 = IntegerDigits[MpData[[jc]], base, intlen2]; ds1 = s1; ds2 = s2; lcs = Length[LongestCommonSubsequence[ds1, ds2]]; If[lcs != intlen1, count++;];]; If[count == nMp, pcount++;Print[s1];];]; Print[pcount];
 2022-11-28, 09:05 #83 Dobri   "ม้าไฟ" May 2018 22×5×23 Posts At the intersection in the Venn diagram of three discrete sets obtained with: - longest common subsequences (LCS); - Ulam clockwise square spiral (UCS); and - binary substring elimination (BSE); there are 9 remaining prime exponents obtained at the intersection of LCS ∩ UCS (file MpLCSUlamX5973Yn2658D1500b28.txt, post #73, https://www.mersenneforum.org/showpo...9&postcount=73) and BSE (up to 9 bits, see posts #81, https://www.mersenneforum.org/showpo...1&postcount=81, and #82, https://www.mersenneforum.org/showpo...0&postcount=82), M137766961, M144753949, M171643919, M174117841, M177777323, M177777421, M185867009, M190145201, and M204871837, as well as 3 remaining prime exponents obtained at the intersection of LCS ∩ UCS (file MpLCSUlamXn5973Y2658D1500b28.txt, post #73, https://www.mersenneforum.org/showpo...9&postcount=73) and BSE (up to 9 bits, see posts #81, https://www.mersenneforum.org/showpo...1&postcount=81, and #82, https://www.mersenneforum.org/showpo...0&postcount=82), M185839957, M185950601, and M185950829. The number of prime exponents in an intersection subset can be increased by: - using bit reversal and cyclic folding for the LCS set; - increasing 𝛥x and 𝛥y for the UCS set; and - using a modified BSE set (by allowing the inclusion of one or more BSE substrings in the prime exponents). At an LCS ∩ UCS ∩ BSE intersection, a local wavefront is formed that can be expanded until the next Mersenne prime (if any) is found. It is like going fishing at specific spots along the river on the basis of empirical observations instead of using a large net to catch all the fish (no prime exponent left behind). Last fiddled with by Dobri on 2022-11-28 at 09:21
2022-11-28, 11:18   #84
kriesel

"TF79LL86GIMPS96gpu17"
Mar 2017
US midwest

163178 Posts

Quote:
 Originally Posted by Dobri there are 9 remaining prime exponents obtained at the intersection of LCS ∩ UCS (file MpLCSUlamX5973Yn2658D1500b28.txt, post #73, https://www.mersenneforum.org/showpo...9&postcount=73) and BSE (up to 9 bits, see posts #81, https://www.mersenneforum.org/showpo...1&postcount=81, and #82, https://www.mersenneforum.org/showpo...0&postcount=82), M137766961, M144753949, M171643919, M174117841, M177777323, M177777421, M185867009, M190145201, and M204871837, as well as 3 remaining prime exponents obtained at the intersection of LCS ∩ UCS (file MpLCSUlamXn5973Y2658D1500b28.txt, post #73, https://www.mersenneforum.org/showpo...9&postcount=73) and BSE (up to 9 bits, see posts #81, https://www.mersenneforum.org/showpo...1&postcount=81, and #82, https://www.mersenneforum.org/showpo...0&postcount=82), M185839957, M185950601, and M185950829.
That's a really subtle difference. It took a while to see it.

2022-11-28, 12:49   #85
Dobri

"ม้าไฟ"
May 2018

1110011002 Posts

Quote:
 Originally Posted by kriesel That's a really subtle difference. It took a while to see it.
Indeed, the letter 'n' in the file names of the attachments to post #73 is used instead of the minus sign '-':
MpLCSUlamX5973Yn2658D1500b28.txt (for y = -2658) and
MpLCSUlamXn5973Y2658D1500b28.txt (for x = -5973).

2023-01-04, 08:54   #86
kriesel

"TF79LL86GIMPS96gpu17"
Mar 2017
US midwest

53·59 Posts

Quote:
 Originally Posted by Dobri At the intersection in the Venn diagram of three discrete sets obtained with: - longest common subsequences (LCS); - Ulam clockwise square spiral (UCS); and - binary substring elimination (BSE); there are 9 remaining prime exponents obtained at the intersection ... M137766961, M144753949, M171643919, M174117841, M177777323, M177777421, M185867009, M190145201, and M204871837, as well as 3 remaining prime exponents obtained at the intersection of ... M185839957, M185950601, and M185950829. ... It is like going fishing at specific spots
Of the 12 above, 3 now have factors found, 9 have prp/proof & successful cert completed. No fish.

2023-01-04, 13:26   #87
Dr Sardonicus

Feb 2017
Nowhere

3×31×67 Posts

Quote:
 Originally Posted by kriesel Of the 12 above, 3 now have factors found, 9 have prp/proof & successful cert completed. No Go fish.
FTFY

2023-01-04, 14:16   #88
kriesel

"TF79LL86GIMPS96gpu17"
Mar 2017
US midwest

11100110011112 Posts

Quote:
 Originally Posted by Dr Sardonicus FTFY
No, I think he's gone fishing plenty already, or at least cast quite widely, fish stories of where might be good, at ~26,500 candidates combined, in the following. (Of which at least ~117 have so far been ruled out.)

30-bit LCS generated, that are not yet factored; ~4939
https://mersenneforum.org/showpost.p...4&postcount=39

28-bit LCS generated, that are not yet factored or verified; ~7128
https://mersenneforum.org/showpost.p...1&postcount=40
https://mersenneforum.org/showpost.p...2&postcount=41

27-bit LCS generated, that are not yet factored or verified; ~8261 (- at least ~42 since verified)
https://mersenneforum.org/showpost.p...5&postcount=42
https://mersenneforum.org/showpost.p...7&postcount=43
https://mersenneforum.org/showpost.p...8&postcount=44

29-bit LCS generated, that are not yet factored; ~5063
https://mersenneforum.org/showpost.p...7&postcount=49

26-bit LCS generated, that are not yet factored or verified; ~450
https://mersenneforum.org/showpost.p...9&postcount=50
some have since been verified with prp/proof gen & CERT or LLDC, ~51 done since the listing was posted

subset of the preceding, that are LCS generated and palindromic prime exponents, 5 unverified or untested, some needing factoring
https://mersenneforum.org/showpost.p...9&postcount=61 of which two now have prp/proof & cert, two have sufficient factoring done

primes nearest 2 coordinates are both factored
https://mersenneforum.org/showpost.p...9&postcount=69

intersection between LCS and Ulam clockwise square spiral sets
https://mersenneforum.org/showpost.p...4&postcount=70
of 5, one lacks a proof upload and cert

intersection subset within x,y +-15000 between LCS generated and UCS generated sets around each central point of 28-bit prime exponents contains several hundred to be tested for the first time.
https://mersenneforum.org/showpost.p...9&postcount=73
~693, of which at least 11 have prp/proof & cert and another 5 found factors

Many of these are < 82589933 so make as good a target as any, on the way to Mp#48-Mp#51* exhaustively verified.

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