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.

According to general Mersenne's, there cannot be any other twin Mersenne primes. It's pretty simple.

What's general Mersenne's?

Oops, I forgot to include an option for infinitely many.

[QUOTE]jinydu What's general Mersenne's? [/QUOTE]

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...

[QUOTE=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...[/QUOTE]

So, how does that show that there are no more twin Mersenne primes?

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.

[QUOTE=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.[/QUOTE]

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

[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.[/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. :ermm:

[QUOTE=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. :ermm:[/QUOTE]

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?

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).

[QUOTE=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).[/QUOTE]

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.