Exponents of Known Mersenne Primes (Numerical Observations) - Page 8

Dobri · 2022-11-22, 06:34

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]]

kriesel · 2022-11-24, 12:33

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?

Dobri · 2022-11-25, 01:06

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) ...

Dobri · 2022-11-25, 01:28

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

Dobri · 2022-11-28, 07:51

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];

Dobri · 2022-11-28, 09:05

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).

kriesel · 2022-11-28, 11:18

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.

Dobri · 2022-11-28, 12:49

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).

kriesel · 2023-01-04, 08:54

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.

Dr Sardonicus · 2023-01-04, 13:26

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

kriesel · 2023-01-04, 14:16

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.

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

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