Exponents of Known Mersenne Primes (Numerical Observations)

[Dobri]

The first post of this thread shows the Hamming distances between the consecutive exponents of known Mersenne primes:

{1,2,1,2,3,1,2,2,3,3,2,6,4,4,5,4,5,5,6,8,4,5,7,5,7,11,11,10,11,10,8,9,9,9,8,11,12,11,11,12,13,11,13,17,11,13,11,14,12,13}.

Below is the Wolfram code used to plot a linear fit for said Hamming distances in the attached PDF file.

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]; hda = ConstantArray[0, nMp - 1];base = 2; intlen = 27; 
ic = 1; While[ic < nMp, ic++; hda[[ic - 1]] = HammingDistance[IntegerDigits[MpData[[ic - 1]], base, intlen], IntegerDigits[MpData[[ic]], base, intlen]];];
Print[hda];
hdafit = LinearModelFit[hda, x, x];
Show[BarChart[hda], Plot[hdafit[x], {x, 1, nMp}], Frame -> True]

[Dobri]

After dividing the Hamming distance by the average number of bits of the pairs of consecutive exponents, the linear fit gives a nearly horizontal line as shown in the attached image.

ClearAll;
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]; hda = ConstantArray[1, nMp - 1]; base = 2; intlen = 27;
ic = 1; While[ic < nMp, ic++; hda[[ic - 1]] = HammingDistance[IntegerDigits[MpData[[ic - 1]], base, intlen], IntegerDigits[MpData[[ic]], base, intlen]]/((Length[IntegerDigits[MpData[[ic - 1]], base]] + Length[IntegerDigits[MpData[[ic]], base]])/2);];
hdafit = LinearModelFit[hda, x, x];
Export["Hamming.jpg", Show[BarChart[hda], Plot[hdafit[x], {x, 1, nMp}], Frame -> True]]

[Dobri]

Let's put the meaning of the graph in post #2 at https://www.mersenneforum.org/showpo...10&postcount=2 into words:

The current estimate of the average base-2 Hamming distance per bit (HAMD/bit) between consecutive exponents of Mersenne primes is bounded in the interval (0.45, 0.5) with a tendency toward 0.5.

[Dobri]

In the attached image, a base-2 HAMD/bit graph for all consecutive primes ≤ 1000003 is shown for a partial comparison with the graph in the previous post #2 at https://mersenneforum.org/showpost.p...10&postcount=2.
The average base-2 HAMD/bit value for all consecutive primes is less than 0.2 and has a downward tendency with the increase of the number of primes included in the computation.

SetDirectory[NotebookDirectory[]]; fname = NotebookDirectory[] <> "HammingPerBitAllPrimes.jpg";
pmax = 1000003; np = PrimePi[pmax]; hda = ConstantArray[0, np-1]; base = 2; intlen = Length[IntegerDigits[pmax, base]];
p1 = 1; p2 = 2; ic = 0; While[p2 < pmax, p1 = p2; p2 = NextPrime[p2]; ic++; hda[[ic]] = HammingDistance[IntegerDigits[p1, base, intlen], IntegerDigits[p2, base, intlen]]/((Length[IntegerDigits[p1, base]] + Length[IntegerDigits[p2, base]])/2);];
hdafit = LinearModelFit[hda, x, x];
Show[BarChart[hda], Plot[hdafit[x], {x, 1, np}], Frame -> True]
Export[fname, Show[BarChart[hda], Plot[hdafit[x], {x, 1, np}], Frame -> True]]

[Dobri]

The dimensionless HAMD/bit term is also called normalized Hamming distance (NHD) which in general describes to what extent two strings are dissimilar.

The tendency for the exponents of Mersenne primes is NHD → 0.5 which corresponds to maximum entropy.

[Dobri]

Obviously, the tendency NHD → 0.5 becomes even more prominent when dividing by the bit length of the larger exponent of a pair of two consecutive exponents (zero padding applies without saying to the smaller exponent in this post and the previous posts) as shown in the attached image.

SetDirectory[NotebookDirectory[]];
fname = NotebookDirectory[] <> "NHD.jpg";
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]; hda = ConstantArray[1, nMp - 1]; base = 2; intlen = 27;
ic = 1; While[ic < nMp, ic++; hda[[ic - 1]] = HammingDistance[IntegerDigits[MpData[[ic - 1]], base, intlen], IntegerDigits[MpData[[ic]], base, intlen]]/Length[IntegerDigits[MpData[[ic]], base]];];
hdafit = LinearModelFit[hda, x, x];
Show[BarChart[hda], Plot[hdafit[x], {x, 1, nMp}], Frame -> True]
Export[fname, Show[BarChart[hda], Plot[hdafit[x], {x, 1, nMp}], Frame -> True]]

[Dobri]

The attached image shows the normalized Levenshtein distance (NLD) (it is obtained as "the minimum number of single-character edits (insertions, deletions or substitutions) required to change one word into the other", see https://en.wikipedia.org/wiki/Levenshtein_distance) between the consecutive exponents of known Mersenne primes.

SetDirectory[NotebookDirectory[]]; fname = NotebookDirectory[] <> "NLD.jpg";
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]; hda = ConstantArray[0, nMp - 1]; base = 2; intlen = Length[IntegerDigits[MpData[[nMp]], base]];
ic = 1; While[ic < nMp, ic++; hda[[ic - 1]] = EditDistance[IntegerDigits[MpData[[ic - 1]], base, intlen], IntegerDigits[MpData[[ic]], base, intlen]]/Length[IntegerDigits[MpData[[ic]], base]];];
hdafit = LinearModelFit[hda, x, x];
Show[BarChart[hda], Plot[hdafit[x], {x, 1, nMp}], Frame -> True]
Export[fname, Show[BarChart[hda], Plot[hdafit[x], {x, 1, nMp}], Frame -> True]]

[Dobri]

The attached image shows the lengths (non-normalized) of the longest common contiguous subsequences (LCS) between the consecutive exponents of known Mersenne primes.

SetDirectory[NotebookDirectory[]]; fname = NotebookDirectory[] <> "LCS.jpg";
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]; hda = ConstantArray[0, nMp - 1]; base = 2;
ic = 1; While[ic < nMp, ic++; intlen1 = Length[IntegerDigits[MpData[[ic - 1]], base]]; intlen2 = Length[IntegerDigits[MpData[[ic]], base]]; hda[[ic - 1]] = Length[LongestCommonSubsequence[IntegerDigits[MpData[[ic - 1]], base, intlen1], IntegerDigits[MpData[[ic]], base, intlen2]]];];
hdafit = LinearModelFit[hda, x, x];
Show[BarChart[hda], Plot[hdafit[x], {x, 1, nMp}], Frame -> True]
Export[fname, Show[BarChart[hda], Plot[hdafit[x], {x, 1, nMp}], Frame -> True]]

[Dobri]

The attached 2 images show the base-2 Smith-Waterman similarity (SWS.jpg, see https://en.wikipedia.org/wiki/Smith%...rman_algorithm, Wolfram function SmithWatermanSimilarity) and the base-2 Needleman-Wunsch similarity (NWS.jpg, see https://en.wikipedia.org/wiki/Needleman%E2%80%93Wunsch_algorithm, Wolfram function NeedlemanWunschSimilarity) between the consecutive exponents of known Mersenne primes.

[Dobri]

The attached 3D image shows the lengths (non-normalized) of the longest common contiguous subsequences (LCS) between all pairs (i, j), i < j, for i, j = 1, 2,..., 51, of the exponents of known Mersenne primes.

SetDirectory[NotebookDirectory[]]; fname = NotebookDirectory[] <> "CLS3D.jpg";
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]; hda = ConstantArray[0, {nMp, nMp}]; base = 2;
ic = 0; While[ic < nMp, ic++; jc = 0; While[jc < nMp, jc++; If[ic < jc,
   intlen1 = Length[IntegerDigits[MpData[[ic]], base]]; intlen2 = Length[IntegerDigits[MpData[[jc]], base]];
   hda[[ic, jc]] = Length[LongestCommonSubsequence[IntegerDigits[MpData[[ic]], base, intlen1], IntegerDigits[MpData[[jc]], base, intlen2]]];
];];];
Show[ListPlot3D[hda, InterpolationOrder -> 0, Filling -> Bottom], Frame -> True]
Export[fname, Show[ListPlot3D[hda, InterpolationOrder -> 0, Filling -> Bottom], Frame -> True]]

[Dobri]

Here is an example of a 15-bit largest common subsequence:

Mersenne exponent #29 = 110503 = 11010111110100111₂, and
Mersenne exponent #39 = 13466917 = 110011010111110100100101₂.