20140418, 00:40  #1 
Apr 2014
5·17 Posts 
Distribution of Mersenne Factors
I've been looking for patterns and such dealing with Mersenne factors, and I found something interesting. I have all the primes less than 1 million along with their multiplicative orders mod 2 in a database, so that's the sample size here. Here's a pattern I found:
The ratio of primes = 7 mod 8 which are Mersenne factors to those primes = 7 mod 8 which are not Mersenne factors ~= 16.3548%. The ratio of primes = 1 mod 8 which are Mersenne factors to those primes = 7 mod 8 which are Mersenne factors ~= 16.3546%. Startlingly close, right? It makes sense that that there are many less primes = 1 mod 8 that are Mersenne factors, since their multiplicative orders aren't guaranteed to be odd like primes = 7 mod 8. But it seems like their distribution might have a solid basis to find. Last fiddled with by tapion64 on 20140418 at 00:44 
20140418, 00:55  #2 
6809 > 6502
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Aug 2003
101×103 Posts
2498_{16} Posts 
Last fiddled with by Uncwilly on 20140418 at 00:56 Reason: Obivously 'in before misc math.' 
20140418, 01:26  #3  
Undefined
"The unspeakable one"
Jun 2006
My evil lair
3·2,027 Posts 
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20140418, 01:42  #4 
Apr 2014
5×17 Posts 
The mods here just love to move stuff to this sub forum. Even though it's relevant to GIMPS. Whatever.
On to the possible mathematical significance. The Sophie Germain primes are distributed evenly along the p = 1 and p = 3 mod 4 lines, and there is a 1:1 correspondence between Sophie Germain primes = 3 mod 4 and Mersenne factors = 7 mod 8. It's not absolutely dead on, but if you go by the approximation for Sophie Germain primes less than n = 2C*n/ln(n)^2, where C is the twin prime constant, and divide it by 2, at 1,000,000 you get ~3458. The number of Mersenne factors = 7 mod 8 less than 1,000,000 is 3217, which is about 7.5% error, which as approximation functions go, isn't that bad for a small n like 1,000,000. Within the space of Mersenne factors, it would appear that primes = 1 mod 8 have the same distribution. Whether that has any meaning... I can't really say right now >.> Last fiddled with by tapion64 on 20140418 at 02:22 
20140418, 01:48  #5  
Apr 2014
85_{10} Posts 
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20140418, 01:51  #6  
"Forget I exist"
Jul 2009
Dumbassville
8384_{10} Posts 
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20140418, 02:05  #7 
Apr 2014
5×17 Posts 
Yeah, I suppose I should've put the mathematical significance in the first post. I was writing it, but then I had to go do something so I left it to edit in later.

20140418, 02:23  #8 
Apr 2014
5·17 Posts 
And now I think I can say. As the primes are evenly distributed across p = 1,3,5,7 mod 8, the normal n/ln(n) divided by 4 is a good approximation for a particular class (e.g. p = 7 mod 8). The ratio of C*n/ln(n)^2 and n/(4*ln(n)) = C*4/ln(n). If we multiply 3458 by this value, we get ~661. The actual value of primes = 1 mod 8 which are Mersenne factors less than 1,000,000 is 629. So if these approximations are valid, then we'd have 4*C^2*n/ln(n)^3 as an approximation for primes = 1 mod 8 that are Mersenne factors less than n.
Since Mersenne factors can only be = 1,7 mod 8, then putting it together, we get C*n*(1+4*C/ln(n))/ln(n)^2 as an approximation for the number of Mersenne factors less than n. This is my conjecture. Last fiddled with by tapion64 on 20140418 at 02:27 
20140418, 02:41  #9  
May 2013
East. Always East.
11010111111_{2} Posts 
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But yeah, the mathy people here don't take kindly to stats. Sure, the 16.355% numbers are interesting, but there's nothing much else to it until we look deeper. It's your line of thinking that's going to uncover something profound that we don't know about numbers, but this isn't the environment to be doing your thinking in. It's kind of too bad that the "some lowly turd like you isn't going to figure something like this out. Someone else would have thought of this by now if it meant anything" mentality is so prevalent around here, but I have to agree that the percentages aren't much to go on... It'll probably take a thousand of these observations before anything meaningful comes out, but it's nice to see you're looking. 

20140418, 03:01  #10 
Undefined
"The unspeakable one"
Jun 2006
My evil lair
3·2,027 Posts 

20140418, 03:07  #11 
Apr 2014
5·17 Posts 
Mawn, something meaningful like the approximation for the number of Mersenne factors less than n, like what I just posted? :p
I'm trying to see how that could link to the distribution of Mersenne primes, but that's a harder link to forge. I just took for example 2^191 and 2^311 and compared the ratio for the number of Mersenne primes less than or equal, I got 1/285.5 and 1/383009, and there's roughly a 4000/3 ratio between those two (and 2^191 is roughly 4000 times 2^311). It looks vaguely promising to pursue this line to narrow down on the form. 
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