20210717, 16:59  #45  
"TF79LL86GIMPS96gpu17"
Mar 2017
US midwest
13443_{8} Posts 
Quote:
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 8digit: 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 8decimaldigit, 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. Last fiddled with by kriesel on 20210717 at 17:49 

20210717, 19:21  #46 
6809 > 6502
"""""""""""""""""""
Aug 2003
101×103 Posts
2×5,059 Posts 

20210717, 20:41  #47  
"刀比日"
May 2018
356_{8} Posts 
Strategies for Manual Testing
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 Quote:
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. 

20210717, 21:07  #48 
Apr 2020
2×3×7×13 Posts 
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? 
20210717, 21:19  #49 
Sep 2002
Database er0rr
2·7·281 Posts 

20210718, 07:38  #50  
Romulan Interpreter
"name field"
Jun 2011
Thailand
2^{4}·613 Posts 
Quote:
Thread closed. Edit: Whhops, sorry, it was closed already. Last fiddled with by LaurV on 20210718 at 07:40 

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