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Old 2014-04-18, 00:40   #1
tapion64
 
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Default 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 2014-04-18 at 00:44
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Old 2014-04-18, 00:55   #2
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Last fiddled with by Uncwilly on 2014-04-18 at 00:56 Reason: Obivously 'in before misc math.'
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Old 2014-04-18, 01:26   #3
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Quote:
Originally Posted by tapion64 View Post
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.
I think you are on to something here and you should spend more of your time to pursue this until you find that pot at the end of the rainbow. We would love to hear more about all the tiniest of minutiae concerning your current direction of exploration into the curious world of numerology. Now if only there were a way we could make use of those figures 16.3548% and 16.3546%.
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Old 2014-04-18, 01:42   #4
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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 2014-04-18 at 02:22
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Old 2014-04-18, 01:48   #5
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Quote:
Originally Posted by retina View Post
I think you are on to something here and you should spend more of your time to pursue this until you find that pot at the end of the rainbow. We would love to hear more about all the tiniest of minutiae concerning your current direction of exploration into the curious world of numerology. Now if only there were a way we could make use of those figures 16.3548% and 16.3546%.
The path that leads to conjectures is filled with many possibly useless figures. You can't give meaning to the figures until you find patterns, and you can't give meaning to the patterns until you relate them to other concepts. Sure, sometimes a pattern is just a coincidence. But sometimes it's not.
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Old 2014-04-18, 01:51   #6
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Quote:
Originally Posted by tapion64 View Post
The mods here just love to move stuff to this sub forum. Even though it's relevant to GIMPS. Whatever.
I think part of that might go back to this reasoning: http://mersenneforum.org/showthread....rum#post358469
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Old 2014-04-18, 02:05   #7
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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.
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Old 2014-04-18, 02:23   #8
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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.

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Old 2014-04-18, 02:41   #9
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Quote:
Originally Posted by retina View Post
I think you are on to something here and you should spend more of your time to pursue this until you find that pot at the end of the rainbow. We would love to hear more about all the tiniest of minutiae concerning your current direction of exploration into the curious world of numerology. Now if only there were a way we could make use of those figures 16.3548% and 16.3546%.
For what it's worth, it was funnier last time when you weren't trying to be funny with it.


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.
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Old 2014-04-18, 03:01   #10
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For what it's worth, it was funnier last time when you weren't trying to be funny with it.
Oh abuse eh? But I came here for an argument.
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Old 2014-04-18, 03:07   #11
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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 2^19-1 and 2^31-1 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 2^19-1 is roughly 4000 times 2^31-1). It looks vaguely promising to pursue this line to narrow down on the form.
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