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 T.Rex 2008-09-16 19:18

[QUOTE=nngs;142758]I assume [TEX]$\log_2(\log_2(M_{\rm n}))$[/TEX] depends on n as [TEX]A+B\times n+C\sin(D\times n+E)[/TEX]

The best fit using all 46 known mersenne primes is:
A=1.1253235 +/- 0.125
B=0.55185022 +/- 0.00484
C=0.68753629 +/- 0.0850
D=0.29490442 +/- 0.00560
E=-3.2489748 +/- 0.149
[/QUOTE]Interesting. How did you find A, B, C, D and E ?

The theoretical (but not yet proved !) Poisson's slope is : [TEX]1/e^\gamma= 0.5614594...[/TEX]
So, your B is not so far.

Tony

 nngs 2008-09-16 20:19

[QUOTE=T.Rex;142782]Interesting. How did you find A, B, C, D and E ?

The theoretical (but not yet proved !) Poisson's slope is : [TEX]1/e^\gamma= 0.5614594...[/TEX]
So, your B is not so far.

Tony[/QUOTE]

I use a [URL="http://www.ittvis.com/ProductServices/IDL.aspx"]IDL[/URL] route called LMFIT. Here is the first sentence in LMFIT help:

[QUOTE]The LMFIT function does a non-linear least squares fit to a function with an arbitrary number of parameters. [/QUOTE]

If I use A+B*n only (which will give me B=0.561...), the residuals show a sinusoidal pattern. So I add a sine term in my guess function.

 petrw1 2008-09-16 22:49

Well, breaking from the mathematically sound 2^n intervals, I charted the number of MPs by 10^n digits:
0 = 7 (for 10^0 digits there are 7 MPs)
1 = 5
2 = 6
3 = 8
4 = 5
5 = 6
6 = 7
7 = 2 (so far)

Again, based on unsound principles, excluding the first interval, I see a pattern (5,6,8,5,6,'7'). So, it would suggest there is one more MP at 10^6 digits. But with all the focus on larger ranges it could be a while before that range clears so I am guessing it will be in the 29M range but not found until November 2009.

 Primeinator 2008-09-17 05:56

I predict M47's exponent will be approximately 47,300,000 and will be discovered in October of next year.

 Uncwilly 2008-09-17 17:59

I have started tabulating the data. I am up to date at this point.

 R. Gerbicz 2008-09-17 18:49

Never. There is no more Mersenne prime, the list is complete.

[quote=Uncwilly;142931]I have started tabulating the data. I am up to date at this point.[/quote]

Once we get a few more, we could make a chart, and send it to mini-geek, so that he can update his post.

"All mersenne primes have been found" is a valid guess, so there should be a bucket for that. In fact, I would say that as of right now, Gerbicz is currently the leading contender in this contest :smile:

 Mini-Geek 2008-09-17 19:25

[quote=uigrad;142935]... and send it to mini-geek, so that he can update his post.[/quote]
Only mods can edit posts after 1 hour, so someone else will have to put it up on the first post, if a chart is made (I think that's a good idea).

 petrw1 2008-09-17 19:28

To the contrary could there be MORE than expected?

I started monitoring the "pretty colorful stats report" almost 3 years ago (Oct 25, 2005).
At the time there were 42 known primes and 2.64 "Expected New Primes" for an expected total between 44 and 45.

Not quite 3 years later we are at 46 known primes and it still reports 1.77 expected for an expected total NOW closer to 48 ... 3 more than expected 3 years ago.

Is this not mathematically significant enough to make one wonder if they are LESS rare than we thought?

 philmoore 2008-09-17 19:50

[QUOTE=petrw1;142937]Is this not mathematically significant enough to make one wonder if they are LESS rare than we thought?[/QUOTE]

Take a look at Chris Caldwell's updated graph at:
[url]http://primes.utm.edu/mersenne/heuristic.html[/url]
Note the last seven points - definitely a significantly tighter clustering than the clusterings at M21 through M23 and M24 through M26. Could be one of the following:

1) Purely a statistical deviation.
2) The beginning of a departure from the heuristic which has been pretty successful so far in characterizing the overall distribution of Mersenne primes.
3) Just the universe's way of encouraging us to continue searching for more Mersenne primes!

 Primeinator 2008-09-17 20:14

[QUOTE]2) The beginning of a departure from the heuristic which has been pretty successful so far in characterizing the overall distribution of Mersenne primes.[/QUOTE]

I'm hoping it is possibility number two. :cool:

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