On Mon, 24 Jan 2000, Paul Landon wrote:
> Subject: Re: Mersenne: Size of largest prime factor
>
> > Pierre Abbat wrote:
> > >If I pick a huge number n at random, how much smaller than n, on average,
> > is
> > >its largest prime factor?
> >
> Jud McCranie wrote:-
>
> > On the average, the largest prime factor of n is n^0.6065, and the second
> > largest is n^0.2117. Reference: Knuth, the Art of Computer Programming,
> > vol 2, section 4.5.4.
> >
> But for Mersennes this might not be the case.
> For the size of exponents that we deal with Mersennes are less
> composite than a random set of ones & zeroes.
> There are many reasons for this, if 2^p-1 has any factors they
> must be bigger than p. They must be +-1 mod 8 etc.
> Looking at the string of ones it certainly has regularity. Indeed
> there is a measure for it, the order of 2 mod 2^p-1 which is very
> low, =p; and any factors have this order as well. This is not
> average.
> This is not new news to most people here, but I have to remind
> myself, it still hasn't been proved whether there are an infinite
> number of Mersenne Primes or an infinite number of Mersenne
> composites.
Erhm?
2^n-1 where n is composite is in itself composite, so showing that there
are infinitely many Mersenne composites is easy. :)
>
> Cheers,
> Paul Landon
--
Henrik Olsen, Dawn Solutions I/S URL=http://www.iaeste.dk/~henrik/
`Can you count, Banjo?' He looked smug. `Yes, miss. On m'fingers, miss.'
`So you can count up to ...?' Susan prompted.
`Thirteen, miss,' said Banjo proudly. Terry Pratchett, Hogfather
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