Hello List:
I might be jumping the gun in here (as I have not read yet all the
Mersenne Digest #574. )
However.....
These
are called
Sophie Germain primes, and it has been proven that there are an
infinite number
of them,
Source:<[EMAIL PROTECTED]>
AND
I'm not sure whether or not it has been proven whether or not there
are
an infinity of Sophie Germain primes of the form 4*n+3. I imagine
there
would be, as there are an infinity of primes in the form 4*n+1 and
4*n+3.
Source:<[EMAIL PROTECTED]>
....stuck to me like a sore thumb.
I am not aware that anyone has yet proven the infinitude of Sophie
Germain Primes. [Granted that, in itself, does not mean anything ;) ]
I am aware of the Hardy-Littlewood conjectures that not only (if proven)
indicate that there are infinite SG Primes but would also give an
approximation as to their frequency (and that info is available on
Professor's Caldwell Prime Pages) but at least until the publication of
Paulo Ribenboim, "The New Book of Prime Number Records,"
Springer Verlag, New York, 1st Ed. -i.e. 1995- this (infinitude of
Sophie Germains and/or infinitude of composite Mersenne Numbers) had
_not_ been proven. I'm also familiar with Lejeune Dirchlelet
demonstration in 1826 of the fact that there are infinite number of
primes in any arithmetical progression with the first term coprime to
the difference. Thus, the infinitude of primes of the forms 4n+3 and
4n+1 is a corollary of this proof.
So, if anyone of the correspondents (or any other member of the list for
that matter) would be so kind as to refer me to the publication (or Web
Page) in which this proof has been announced, I would be eternally
grateful.
Now, as a disclaimer, I am speaking about "mathematical proof' on the
Godfrey H. Hardy tradition, and not merely heuristical approaches.
Thanks,
Rodolfo Ruiz
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