> 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.
Cheers,
Paul Landon
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