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 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

 2004-05-31, 10:12 #1 jinydu     Dec 2003 Hopefully Near M48 2·3·293 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.
 2004-05-31, 10:44 #2 TTn   34×107 Posts According to general Mersenne's, there cannot be any other twin Mersenne primes. It's pretty simple.
 2004-06-01, 02:16 #3 jinydu     Dec 2003 Hopefully Near M48 2×3×293 Posts What's general Mersenne's? Oops, I forgot to include an option for infinitely many.
2004-06-02, 03:07   #4
TTn

2·7·13·43 Posts

Quote:
 jinydu What's general Mersenne's?
Riesel primes(k*2^n-1), with k<2^n.
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...

2004-06-02, 04:21   #5
jinydu

Dec 2003
Hopefully Near M48

33368 Posts

Quote:
 Originally Posted by TTn Riesel primes(k*2^n-1), with k<2^n. 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...
So, how does that show that there are no more twin Mersenne primes?

 2004-06-02, 04:42 #6 ixfd64 Bemusing Prompter     "Danny" Dec 2002 California 2·32·137 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.
2004-06-02, 05:17   #7
jinydu

Dec 2003
Hopefully Near M48

2×3×293 Posts

Quote:
 Originally Posted by ixfd64 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.
So that doesn't really prove that there are no further Mersenne primes. After all, a lot of things can happen between 10^2,598,000 and infinity.

2004-06-02, 09:04   #8
TTn

31×37 Posts

Quote:
 So that doesn't really prove that there are no further Mersenne primes. After all, a lot of things can happen between 10^2,598,000 and infinity.

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.

2004-06-02, 13:35   #9
R.D. Silverman

Nov 2003

164448 Posts

Quote:
 Originally Posted by TTn 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.
Hey guys. Everyone, and I do mean EVERYONE would be a lot better off if
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?

 2004-06-02, 14:19 #10 jinydu     Dec 2003 Hopefully Near M48 33368 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 2004-06-02 at 14:20
2004-06-02, 15:32   #11
R.D. Silverman

Nov 2003

22·5·373 Posts

Quote:
 Originally Posted by jinydu 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).
They don't need to read through all the relevant literature. They DO need to
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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