20210716, 17:36  #34 
"刀比日"
May 2018
239 Posts 

20210716, 17:58  #35 
Undefined
"The unspeakable one"
Jun 2006
My evil lair
2·23·137 Posts 
Digit sums are awesome and have predictive power. Yes they do!
I have discovered, after extensive and exhaustive (and exhausting) analysis of the base2 digit sums of all the Mersenne primes that we know of today^{*}, that ALL the digit sums for EVERY MP have a special pattern whereby they have no other divisors other than 1 or themselves.
OMG, why has no one thought of this before!!!!!!! That must say something, right? Am I awesome or what! ^{*} Try saying that quickly in a single breath. Phew. Last fiddled with by retina on 20210716 at 18:05 
20210716, 18:08  #36  
"TF79LL86GIMPS96gpu17"
Mar 2017
US midwest
17×349 Posts 
Quote:
Last fiddled with by kriesel on 20210716 at 18:19 Reason: (bold portion subsequently removed by retina, which makes it a whole other kettle o' fish)) 

20210716, 18:10  #37 
Undefined
"The unspeakable one"
Jun 2006
My evil lair
2·23·137 Posts 
I got too excited from my awesome discovery and had some extra nonsense in there. I already edited it.
Last fiddled with by retina on 20210716 at 18:10 
20210716, 18:37  #38 
Feb 2016
! North_America
130_{8} Posts 
But can you predict M51 from the first 50? I fail to see how any digitsums are siginificant, and (maybe) higher numbers tend to have higher digitsums, since the later part looks random and it evens out, but the earlier part tends to rise steadily. (graph not related)
(was unable to use the data directly, too much for spreadsheet graph to handle) Last fiddled with by thyw on 20210716 at 18:58 
20210717, 12:13  #39 
"刀比日"
May 2018
239 Posts 
Digit Sum Distributions of MersenneNumber Exponents
The attached image contains the plots of:
 The digit sum distribution of known Mersenneprime exponents (green dots) in the interval [2, 47];  The digit sum distribution of 8digit Mersennenumber exponents (red dots) in the interval [2, 71]; and  The digit sum distribution of 9digit Mersennenumber exponents (blue dots) in the interval [2, 80]. The three distinct distributions are normalized to unity with respect to:  The total number of 51 known Mersenneprime exponents (Mp);  The total number of 5,761,455 8digit Mersennenumber exponents (Mn); and  The total number of 50,847,534 9digit Mersennenumber exponents (Mn). Last fiddled with by Dobri on 20210717 at 13:10 
20210717, 12:24  #40  
"刀比日"
May 2018
239 Posts 
Quote:
Assuming that there are several Mp in the 9digit range, one would expect that at least one is among the known digit sums (and probably among the digit sums with the highest frequency of occurrence, like 38, 41, etc.). Note that small numbers also have high digit sums if the Mn exponent contains many '8' and '9' digits. The peculiar thing is that the 51 known Mp have digit sums mainly in the lower half of the Mn digit sum distributions. Last fiddled with by Dobri on 20210717 at 12:25 

20210717, 14:44  #41 
6809 > 6502
"""""""""""""""""""
Aug 2003
101×103 Posts
13·19·41 Posts 

20210717, 15:02  #42  
"TF79LL86GIMPS96gpu17"
Mar 2017
US midwest
17·349 Posts 
Quote:
Quote:
sum ~ 5 + (d2)*4.5 + 5 = 10 + 4.5 * (d2) = 1 + 4.5 d So, 8 digits, ~37; 7 digits, ~32.5; 9 digits, ~41.5. Minimum digit sum, ~2; Maximum ~9d; subject to the constraint that repdigits can not be prime unless their length is prime, and digits one, so for multiple digits, maximum sum is slightly lower, for a nearrepdigit containing mostly digits of value base1. For example, base ten again, max prime exponent p < 10^{9} = 999999937 not 999999999; the next smaller prime exponent 999999929 is a slightly higher digit sum and a nearrepdigit. There are other nearrepdigit primes with slightly higher digit sums. About a million lower, there's a nearrepdigit 998999999 which exponent is prime, providing the max possible digit sum in base ten 9 digit primes, 80. I note there's still not much of an answer re why base 10 digits. "Convenience" works for me.Indeed. I was just thinking of using that. Last fiddled with by kriesel on 20210717 at 15:37 

20210717, 16:21  #43  
"刀比日"
May 2018
239 Posts 
Quote:
For instance, attached are the base2 distributions. In the second image, the base10 distributions are replotted to connect the dots in the graphs with lines. 

20210717, 16:58  #44 
Apr 2020
1044_{8} Posts 
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 noone except perhaps Dobri.
Feels like it's about time for a mod to close the thread. 
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