
View Poll Results: How many other Twin Mersenne Primes are there besides the three mentioned below?  
0  22  84.62%  
1  1  3.85%  
2  0  0%  
More than 2, but finitely many  3  11.54%  
Voters: 26. You may not vote on this poll 

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20040531, 10:12  #1 
Dec 2003
Hopefully Near M48
3336_{8} Posts 
Twin Mersenne Primes
There are 3 known "Twin Mersenne Primes":
M3 and M5 M5 and M7 M17 and M19 More precisely, if both M(p) and M(p+2) are both prime, then they are called Twin Mersenne Primes. 
20040531, 10:44  #2 
4,943 Posts 
According to general Mersenne's, there cannot be any other twin Mersenne primes. It's pretty simple.

20040601, 02:16  #3 
Dec 2003
Hopefully Near M48
2×3×293 Posts 
What's general Mersenne's?
Oops, I forgot to include an option for infinitely many. 
20040602, 03:07  #4  
2^{2}·1,423 Posts 
Quote:
3 7 11 23 31 47 79 127 191 223 239 383 479 607 863 1087 1151 1279... With general Mersenne primes the expansion of the exponent n, is not just about being prime. It's about being the largest exponent yet! gM =3 7 31 127 1279 3583 5119 6143 8191 ... n = 2 3 5 7 8 9 10 11 13... 

20040602, 04:21  #5  
Dec 2003
Hopefully Near M48
2×3×293 Posts 
Quote:


20040602, 04:42  #6 
Bemusing Prompter
"Danny"
Dec 2002
California
2·1,237 Posts 
It's like a paradox. Twin Mersenne primes would become extremely rare if they even exist after M19, but then again, numbers go on infinetely.

20040602, 05:17  #7  
Dec 2003
Hopefully Near M48
3336_{8} Posts 
Quote:


20040602, 09:04  #8  
2^{4}·3·151 Posts 
Quote:
You almost have a point, in that technically all properties of numbers lay beyond mankinds boundries, ie 99.999...% of them.(Guys law of numbers) But If there is a known mod, then we can safely say that there can be no other primes. This is the case as is proven with RMA. I'll break it down in my next message. 

20040602, 13:35  #9  
"Bob Silverman"
Nov 2003
North of Boston
2^{3}·3·311 Posts 
Quote:
they actually took some time to READ about this subject, rather than indulge in random (and mostly wrong) speculation. While a proof is lacking there are very good reasons for believing that the number of twin Mersenne primes is finite and furthermore that we already know all of them. There are strong heuristic arguments which suggest that the number of Mersenne primes between [2^n and 2^2n] is exp(gamma), independent of n as n>oo. The arguments strongly suggest that the Mersenne primes M_p have a Poisson distribution with respect to lg p (log base 2). If this is indeed the case, then there will be finitely many Mersenne twins. There can be no more than O(sum(1/log(M_p) * 1/log(M_p+2))) and this sum CONVERGES over p, let alone over twin prime pairs (p, p+2). The expected number of twin Mersenne primes that are unknown is no more than sum (p > 12,441,000 1/log(M_p) 1/log(M_p+2) ) where (p,p+2) runs over twin prime pairs. This sum is VERY small. 12,441,000 is the bound below which we are sure there are no more Mersenne primes. Now can we put these silly arguments to rest? 

20040602, 14:19  #10 
Dec 2003
Hopefully Near M48
2·3·293 Posts 
In my opinion, this forum would have a lot to lose if users could only post their mathematical ideas after having read through all the relevant literature, mainly because I think this forum should be open to people of all levels of mathematical knowledge. I don't think there is any real harm done in speculating (even incorrectly), especially because the speculator may learn something. After all, papers are a method for people to communicate theorems, conjectures and ideas; and Internet forums are just another method (albeit less formal and rigorously reviewed).
Furthermore, I do not think that mathematical problems should be put to rest (at least not fully) until they have been rigorously settled with proofs. Aside from the certainty they provide, these proofs often provide new insights and lead to further interesting questions (as in the case of Fermat's Last Theorem). Last fiddled with by jinydu on 20040602 at 14:20 
20040602, 15:32  #11  
"Bob Silverman"
Nov 2003
North of Boston
2^{3}·3·311 Posts 
Quote:
at least read the basics of number theory; i.e. a first course book. They also need to learn how to correctly pose their questions. One can't even begin to talk about whether there are infinitely many Mersenne primes without at least knowing basic probability and the definition of a probability density function. One can't properly discuss a subject without knowing the language. 

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