20040517, 14:11  #56  
5,857 Posts 
Quote:
So therefore I claim that your limits are true, but the correct number is not 3/2, but rather SQRT(2). Let's call it WhateverIsYourName  Motl's theorem, if no one else knows it. ;) 

20040517, 14:31  #57  
6809 > 6502
"""""""""""""""""""
Aug 2003
101×103 Posts
10010001111101_{2} Posts 
You folks may want to look at http://www.utm.edu/research/primes/n...tMersenne.html
Quoting from that site: (emphasis mine) Quote:


20040517, 17:48  #58  
"Patrik Johansson"
Aug 2002
Uppsala, Sweden
651_{8} Posts 
Quote:
Code:
Factoring only : 8312 Factored composite : 13186 LucasLehmer testing : 62446 LucasLehmer composite: 74660 Doublechecking LL : 14131 Doublechecked LL : 39640 Prime, VERIFIED : 1 Prime, UNVERIFIED : 1     TOTAL : 84889 TOTAL : 127488 Quote:


20040517, 18:41  #59 
Aug 2003
Upstate NY, USA
2×163 Posts 
maybe the reason he didn't get an email has to do with the Perl issues on the server lately
but he found out soon enough anyways and we'll all know in about a month  sooooo far away 
20040518, 02:30  #60 
Dec 2003
Hopefully Near M48
2·3·293 Posts 
Does anyone know where I can find an applet that calculates Li(x) (logarithmic integral, as in the Prime Number Theorem) for VERY large x (as large as the numbers we're testing for primality)?

20040518, 03:00  #61  
Dec 2003
Hopefully Near M48
1758_{10} Posts 
Quote:
Now, pi(8,000,000) = 539,777 This means that there are 539,777 Mersenne numbers that could be prime, of which only 38 are actually prime. The density is thus about 7.04 * 10^5. By comparison Li(M(8,000,000)) / M(8,000,000) ~= 1.80 * 10^7. This means that Mersenne numbers are about 391 (not 2) times more likely to be prime than "ordinary" numbers, at least when n < 2^(8 million). However, it should be noted that there is significant error in that estimate because so few Mersenne primes are known (i.e. 38 is a small number, so there is still plenty of room for statistical uncertainty). For instance, the density of Mersenne primes up to M6972592 is significantly different from the density up to M6972593 (i.e. a single Mersenne prime changes the density heavily while a single "ordinary" prime has little effect on the density). Still, I would be quite confident to say that Mersenne numbers less than M(8,000,000) are hundreds of times more likely to prime than ordinary natural numbers less than M(8,000,000) I think the reason for this is probably because Mersenne numbers have less possible factors than "ordinary" numbers of similar size. Last fiddled with by jinydu on 20040518 at 03:12 

20040518, 04:15  #62 
Sep 2002
Vienna, Austria
3·73 Posts 
Could anyone post the 4 candidate exponents along with their residue?

20040518, 04:15  #63 
Dec 2003
Hopefully Near M48
6DE_{16} Posts 
Just for comparison, I'll make a table.
Let r = (density of Mersenne primes) / (density of regular primes), where (density of Mersenne primes) = (# of Mersenne primes) / (# of prime exponents). Let n = Mersenne exponent nMersenne prime densityRegular prime densityr 101.000.1685.95 1000.4001.46*10^227.3 10008.33*10^21.44*10^357.7 100001.79*10^21.44*10^4124 10^52.92*10^31.44*10^5202 10^64.20*10^41.44*10^6291 8*10^67.04*10^51.80*10^7390 This table clearly seems to show that Mersenne numbers have an "advantage" over regular numbers, and that this advantage gets larger as the numbers get larger. This means that Mersenne primes do appear to thin out (as everyone probably knows), but not as quickly as regular primes. In case anyone wants to know, I assumed that Regular prime density = Li (2^n) = (1/ (n*ln 2)) + (1/ (n*ln 2)^2) + (2/ (n*ln 2)^3). I got that formula from http://mathworld.wolfram.com/PrimeNumberTheorem.html. DoubleCheckers: Work hard, and I can add another row to that table! NOTE: Sorry about the dashes. Its the only way I could make the table display correctly (at least on my browser). Last fiddled with by jinydu on 20040518 at 04:28 
20040518, 06:33  #64 
2^{4}·83 Posts 
PD versus OD
A statistician, would be looking at the least sum of squares.
You know that line between predicted data, and observed data. :surprised This feature is built into RMA, as an accessory! 
20040518, 09:00  #65  
Dec 2003
Hopefully Near M48
11011011110_{2} Posts 
Quote:
EDIT: A quadratic fit works very well. x = Log (base 10) n y = r y = 9.0921x^2  6.3465x + 2.0573, with an R^2 value of 0.9994 (according to Microsoft Excel). If you prefer natural logarithms: x = ln (n) y = r y = 1.7147x^2  2.7543x + 2.0516, with an R^2 value of 0.9994 That's a good fit considering the small sample size of only 38 Mersenne primes. Last fiddled with by jinydu on 20040518 at 09:13 

20040518, 09:42  #66 
2^{3}×5^{2}×47 Posts 
Ah... hem I was refering to:
http://www.utm.edu/research/primes/n...tMersenne.html The exact line between these lines, providing it is still a candidate ofcourse. Or more exactly, the line between general Mersenne primes. (Download RMA for details at:) http://www.15k.org/rma/ Shane F. TTn 
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