Re: More primality testing (was: Elementary group theory: Proof of Fermat-Maas ...)

In article <877iv3p69g.fsf@xxxxxxxxxxxxxxxxxxxx>,
Phil Carmody <thefatphil_demunged@xxxxxxxxxxx> wrote:

"*** T. Winter" <***.Winter@xxxxxx> writes:
In article <877iv4r6yg.fsf@xxxxxxxxxxxxxxxxxxxx> Phil Carmody
<thefatphil_demunged@xxxxxxxxxxx> writes:
> "*** T. Winter" <***.Winter@xxxxxx> writes:
> > As far as I know there are no prizes for proving a number prime.
>
> FYI, there are. the first proven primes over 10^7 million,
> 10^8 million, and 10^9 million digits will each receive
> a prize from a group of bodies, I forget precisely whom.
> GIMPS is very close to the 10^7 record, having churned
> out several fairly close to that size.

Oh, well, never too old to learn. But in that case determined ways are the
best. And Mersenne primes are the way to go, because those are the easiest
to prove prime.

Are they easier to prove than Generalised Fermat Numbers? Or than
Generalised Eisenstein Fermat Numbers? Please explain quantitatively
how.

I don't believe they're the way to go at all, because the candidates
grow too quickly (complete with a decrease in the density), and they
cannot be effectively sieved. Every individual number must be trial-
factored before testing. If you want to attack those targets then use
a family which won't end up being twice as large after a hundred
thousand failed tests.

You may be right - you know more about these things than I do - but
is it not the case that for several hundred years now the title of
Largest Known Prime has been held by an uninterrupted succession of
Mersenne primes?

--
Gerry Myerson (gerry@xxxxxxxxxxxxxxx) (i -> u for email)
.