Digit sum of a Mersenne-prime exponent

[kriesel]

Originally Posted by Dobri

The peculiar thing is that the 51 known Mp have digit sums mainly in the lower half of the Mn digit sum distributions.

I think there is an analog to or operation of Benford's law here. If Mersenne primes occur on the average, ~1.476:1 spacing (https://primes.utm.edu/mersenne/heuristic.html) on exponent, any n:1 exponent ratio with n large compared to 1.476 will be probability weighted more toward the lower left digit values, lower second-from-left, etc. Uniformly in log space is nonuniformly, & favorable to lower left digits, in linear space. https://www.mersenneforum.org/showpo...35&postcount=6
https://primes.utm.edu/notes/faq/NextMersenne.html
https://en.wikipedia.org/wiki/Benford%27s_law

For separate digit lengths of exponents in base 10:
1 digit: 2 3 5 7; 7 is as good as it gets since 8 and 9 are composite; average 4.25
2 digit: 13 17 19 31 61 89; 89 is maximal possible sum 17; average per digit sum in 4 8 10 4 7 17; sum 50 / 6 = 8.33 / 2 digits = 4.167/digit
3 digit: 107 127 521 607; 8 10 8 13; 39 / 4 = 9.75; 3.25/digit
4 digit: 1279 2203 2281 3217 4253 4423 9689 9941; 19 7 13 13 14 13 32 23; 134 / 8 = 16.75; 4.1875/digit
5 digit: 11213 19937 21701 23209 44497 86243; 8 29 11 16 28 23; 95 / 6 = 15.833; 3.167/digit
6 digit: 110503 132049 216091 756839 859433; 10 19 19 38 32; 118 / 5 = 23.6; 3.933/digit
7 digit: 1257787 1398269 2976221 3021377 6972593; 37 38 29 23 41; 168 / 5 = 33.6; 4.8/digit
8-digit:
exponent digitsum
13466917 37
20996011 28
24036583 31
25964951 41
30402457 25
32582657 38
37156667 41
42643801 28
43112609 26
57885161 41
74207281 31
77232917 38
82589933 47
sum 452 / 13 exponent = 34.769 / 8 digits = 4.346/digit
(apologies for any lingering math errors)

So for 8-decimal-digit, the histogram of # of Mp vs. digitsum value is
25 1
26 1
28 2
31 2
37 1
38 2
41 3
47 1
Graphing that manually in black with only digitization noise +-0.5 counts atop Dobri's base 10 digit sum distributions, using the existing rulings for scale for convenience yields the attachment. The statistical sample size is terribly small.

[Uncwilly]

Originally Posted by Dobri

It is not a matter of convenience. Simply there is no rush to spill the beans in a single post.

So you have been intending to stretch this out. We can put an end to this very quick.

Thread closed

[Dobri]

Closing the thread at
https://mersenneforum.org/showthread.php?t=26997
was premature because the OP was not given a chance to respond, especially at a stage when a post was submitted in their favor.

The post at
https://mersenneforum.org/showpost.p...3&postcount=44

Originally Posted by charybdis

Here's a comparison between the expected number of primes with each digit sum according to the LPW heuristic and the actual number observed, up to the current first testing limit of p=103580003. Doesn't look biased towards low digit sums, which I'm sure will come as a surprise to no-one except perhaps Dobri.

Feels like it's about time for a mod to close the thread.

contains an image showing "the expected number of primes with each digit sum according to the LPW heuristic".

Therefore, it appears one could select exponents for manual testing for a given digit sum as often as indicated by the corresponding expectancy in accordance with the LPW heuristic.

Therefore, the OP would like to respectfully request the previous thread to be reopened and/or the discussion to be continued in this thread instead.

[charybdis]

You have misunderstood the point of my post. It was not intended to support your views. All I did was calculate the probabilities given by the LPW heuristic for each prime up to 103M and add up the probabilities for each digit sum. My graph therefore shows what we expect if digit sum has no effect on the likelihood of being prime - and the actual distribution of primes is consistent with this.

5 is the digit sum with the highest expected number of primes because there are several very small primes with digit sum 5, namely 5, 23 and 41, and the heuristic gives high probabilities for these. For p=5 it gives the nonsensical probability of 1.748. This does not make higher exponents with digit sum 5 more likely to be prime!

Please can a mod close this thread too?

[paulunderwood]

Originally Posted by charybdis

Please can a mod close this thread too?

Topic exhausted. Enuf said.

[LaurV]

Originally Posted by Dobri

This is an early attempt to eventually improve the competitiveness of Mersenneries with modest resources.
I am still figuring out to what extent the combination of digit sum histograms obtained for multiple prime bases could help.

What a big collection of great, empty words, and what a waste of time and computing resources...
Thread closed.

Edit: Whhops, sorry, it was closed already.

[Dobri]

The minimal bases b_min for which the digit sums of the prime exponents of the known Mersenne primes are distinct (different from each other) is given in the following list:

b_min = {2, 2, 3, 7, 7, 10, 10, 20, 20, 21, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 82, 82, 82, 82, 108, 108, 108, 108, 108, 108, 108, 108, 108, 108, 108, 108, 108, 108, 110, 110, 110, 110, 110, 110, 142, 142}

for n = 1, 2, …, 51.

Therefore, b_min = 142 for n = 51, and the distinct base-142 digit sums for the 51 prime exponents of known Mersenne primes are

{2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 98, 43, 10, 88, 25, 115, 164, 52, 101, 71, 215, 197, 128, 85, 82, 92, 100, 214, 220, 233, 179, 208, 254, 134, 170, 284, 289, 124, 172, 224, 319, 236, 347, 325, 167, 290, 250, 308, 311}.

If one assumes for manual selection that b_min remains equal to 142 also for n = 52, then prime exponents for finding M52 (if any) could be selected for further testing if b_min = 142.

(* 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};
nMpmax = Length[MpData]; bm = ConstantArray[0, nMpmax]; basemax = 1000; nMp = 0; While[nMp < nMpmax, nMp++; ds = ConstantArray[0, nMp]; check = 0; base = 1;  While[(base < basemax) && (check == 0), base++; sum = 0; ic = 0; While[ic < nMp, ic++; ds[[ic]] = Total[IntegerDigits[MpData[[ic]], base]];]; dssort = Sort[ds]; check = 1; ic = 1; While[ic < nMp, ic++; If[(dssort[[ic]] == dssort[[ic - 1]]), check = 0;];]; If[check == 1, bm[[nMp]] = base;];];];
Print[bm]; Print["b_min = ", base]; Print[ds];

Beyond b_min = 142, larger base-b values (for which the digit sums of the prime exponents of the known Mersenne primes are distinct) are

b = {224, 230, 278, 300, 314, 330, 334, 336, 342, 352, 374, 398, 402, 404, 412, 414, 418, 424, 432, 444, 447, 450, 458, 459, 467, 468, 473, 474, 488, 490, 498, 504, 507, 512, 518, 522, 541, 543, 545, 546, 548, 552, 555, 566, 572, 576, 584, 585, 588, 594, 608, 614, 620, 627, 630, 642, 654, 656, 658, 660, 674, 682, 683, 688, 690, 696, 699, 704, 706, 713, 718, 722, 723, 726, 738, 740, 742, 746, 750, 756, 762, 768, 770, 777, 778, 793, 797, 798, 800, 811, 812, 814, 816, 820, 826, 827, 830, 833, 835, 836, 847, 852, 854, 857, 863, 866, 868, 870, 878, 880, 882, 884, 887, 890, 894, 896, 897, 900, 905, 906, 910, 914, 922, 923, 924, 926, 929, 930, 933, 935, 936, 940, 941, 942, 944, 952, 954, 958, 960, 962, 974, 975, 978, 979, 980, 982, 984, 988, 993, 994, 996, 998, 1000,...}.