General formula

[pacionet]

Is it possible that a general formula exist for all Mersenne primes ?

[ewmayer]

If you're talking about all that are known *and* those yet to be found, then it's exceedingly improbable. Finding a formula that yields all that are already known (say a smooth function f(x) such that f(n) gives the exponent of the (n)th Mersenne prime) is of course a (rather silly) exercise in curve fitting.

[Orgasmic Troll]

of course there is. M(n) where M(n) is the n-th Mersenne prime. if you don't mind filtering through results, you can even write M'(n) where M'(n) = 2^n-1. I guarantee you that every mersenne prime will be listed, you just have to disregard the composite mersenne numbers

[pacionet]

Originally Posted by TravisT

of course there is. M(n) where M(n) is the n-th Mersenne prime. if you don't mind filtering through results, you can even write M'(n) where M'(n) = 2^n-1. I guarantee you that every mersenne prime will be listed, you just have to disregard the composite mersenne numbers

I mean , of course, a formula f(n) where , when n varies , f(n) returns a Mersenne prime number (for all n values)

[R.D. Silverman]

Originally Posted by pacionet

I mean , of course, a formula f(n) where , when n varies , f(n) returns a Mersenne prime number (for all n values)

Yes. Such a formula exists. But it is *useless* for computational
purposes.

I will give a hint how to construct such a formula.

Let the n'th Mersenne prime be given as M(n) = 2^(p_n) - 1
where p_n is the exponent of the n'th prime.

Consider the constant:

alpha = sum(i=1 to oo) of 10^(-2i) p_i. This is a well defined
real number. The sum clearly converges.

Now, given this constant, one can compute M(n) by multiplying
alpha by 10^(2n), extracting the fractional part
of 10^(2n) alpha, subtracting then truncating the part of the fraction after the trailing 0's that follow p_n. One can do this with a suitable
combination of floor functions and simple multiplications that I am too
lazy to work out at the moment. A similar formula may be found in
Hardy and Wright except it gives the n'th prime, instead of the n'th
Mersenne prime.


It is useless, because we have no way of computing alpha to find
as yet unknown primes.

But the formula does indeed *exist* because alpha exists.

[sghodeif]

of course there is.2^F-1 where F one of Fermates prime numbers(in special form) .

[akruppa]

sghodeif, what exactly are you referring to, what precisely do you mean by "special form" and how do Fermat numbers enter the picture?

Alex

[ewmayer]

Originally Posted by akruppa

sghodeif, what exactly are you referring to, what precisely do you mean by "special form" and how do Fermat numbers enter the picture?

sghodeif is simply trying very hard to get this thread moved to the "miscellaneous math threads forum," Alex.

[Citrix]

I think he is telling us that he needs coffee, as the icon says.

Citrix

[akruppa]

Well, lets not dismiss it without at least giving him a chance to explain himself.

Alex

[Citrix]

He might be saying that

2^(2^2^x)+1)-1 is always prime, for prime fermat's.

So 2^3-1 is prime
2^5-1 is prime
2^17-1 is prime
but 2^257-1 is not prime?

So still not sure.