20210713, 17:11  #1 
"刀比日"
May 2018
13×19 Posts 
Digit sum of a Mersenneprime exponent
The base10 digit sums of the Mersenne exponents for the 51 known Mersenne primes are as follows:
2, 3, 5, 7, 4, 8, 10, 4, 7, 17, 8, 10, 8, 13, 19, 7, 13, 13, 14, 13, 32, 23, 8, 29, 11, 16, 28, 23, 10, 19, 19, 38, 32, 37, 38, 29, 23, 41, 37, 28, 31, 41, 25, 38, 41, 28, 26, 41, 31, 38, 47. An empirical observation shows that the digit sums are either prime numbers or have only 1 or 2 prime factors where the second factor is always 2, for instance: 25 = 5^2, 32 = 2^5, 28 = 2^2 x 7, etc. 
20210713, 18:03  #2 
Apr 2020
11·53 Posts 
This is a meaningless observation, for the same reason that it's meaningless to note that the digit sums are all below 50.
Digit sums of Mersenne prime exponents (apart from 3 itself) can't be divisible by 3, since then the exponent would also be divisible by 3 and therefore not prime. Taking that into account, the smallest possible digit sums which don't satisfy your condition are 35, 55, 65, 70, 77. Given the size of the Mersenne exponents searched so far, all of these apart from 35 would be unlikely to have appeared. It shouldn't be a massive surprise that 35 hasn't come up yet. After all, neither has 34. In the long run, most numbers don't satisfy your condition, so eventually we should expect most exponents of Mersenne primes not to satisfy it either. 
20210713, 22:04  #3 
"刀比日"
May 2018
13×19 Posts 
Applied statistics is not just a science but also an art. When one attempts to estimate what would be the next Mersenne prime on the basis of the current small sample size of known Mersenne primes, it depends what are the risks a Mersennery player could take.
Players having a large number of fast computers would not hesitate to include exponents with digit sums equal to 35, 55, etc. Players with a small number of modest computers could look at empirical observations to eliminate less probable choices for manual testing such as 35. Also, the digit sum histogram of all possible digit sums for the entire testing range has a maximum at around 41. While digit sums 38 and 41 appear frequently in the list of discovered Mersenne primes, the digit sum 40 never occurred and it is a less probable choice for manual testing. Therefore, Mersenneries with modest resources could seek a tradeoff between risks to be taken for manual testing and available computational resources. 
20210713, 22:15  #4  
Undefined
"The unspeakable one"
Jun 2006
My evil lair
2^{2}·3·17·31 Posts 
Quote:
What if the reverse Gambler's fallacy is the case here? You should then be choosing digit sums that are not in the above set because those outputs are already "used up" so different values will be more likely, right? How would you know which position to choose? But, whatever the case, it doesn't matter what you choose as long as you have something to reach for. Last fiddled with by retina on 20210713 at 22:16 

20210713, 22:35  #5  
"刀比日"
May 2018
13·19 Posts 
Quote:
One could go wild and select digit sum 35 for manual testing, but preferably not if the test on a modest computer would take a year. 

20210713, 22:35  #6  
"Rashid Naimi"
Oct 2015
Remote to Here/There
5·439 Posts 
Spurious Correlations
Quote:
Last fiddled with by a1call on 20210713 at 22:38 

20210714, 00:53  #7 
"TF79LL86GIMPS96gpu17"
Mar 2017
US midwest
2^{3}·761 Posts 
In that vein:
Number of GIMPS Mersenne prime discoveries versus exponent digit count 16 digits: 0 7 digits: 4 8 digits: 13 9 digits or higher: 0 So by that spurious measure, we should keep testing 7 digit and mostly 8 digit exponents? Perhaps move this thread to Misc. Math? Last fiddled with by kriesel on 20210714 at 00:56 
20210714, 01:36  #8  
Apr 2020
1001000111_{2} Posts 
Quote:
Code:
gp > c=exp(Euler());prob=1;forprime(p=3,103580003,if(sumdigits(p)==35,if(p%4==1,pprob=1c*log(6*p)/(p*log(2));prob*=pprob,pprob=1c*log(2*p)/(p*log(2));prob*=pprob)));print(prob) 0.29152692322560544242179474578006423121 For digit sum 40, we get Code:
gp > c=exp(Euler());prob=1;forprime(p=3,103580003,if(sumdigits(p)==40,if(p%4==1,pprob=1c*log(6*p)/(p*log(2));prob*=pprob,pprob=1c*log(2*p)/(p*log(2));prob*=pprob)));print(prob) 0.48796250668372976034521929359700885331 

20210714, 03:37  #9  
"刀比日"
May 2018
13·19 Posts 
Quote:
There is an alternative interpretation for the graph of log2(log2(Mn)) versus n. Perhaps another line with a smaller slope is formed after M40 or even the graph starts to reach saturation that could indicate that there is no M52 at all. 

20210714, 04:18  #10 
"刀比日"
May 2018
11110111_{2} Posts 

20210714, 12:23  #11  
Apr 2020
11×53 Posts 
Quote:
Quote:
And what on earth is "the graph starts to reach saturation" supposed to mean? What object is getting saturated in order to prevent any more Mersenne primes existing? For what it's worth, even if Mersenne numbers were no more likely to be prime than random numbers of their size, the Prime Number Theorem would still tell us that there should be infinitely many Mersenne primes. 

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