Re: Help on 2^(big)-1
- From: klewis@xxxxxxxxxxxxxxx (Keith A. Lewis)
- Date: Wed, 30 Nov 2005 18:51:53 +0000 (UTC)
israel@xxxxxxxxxxx (Robert Israel) writes in article <dmjhl3$ge8$1@xxxxxxxxxxxxxxxxxxxxxx> dated 30 Nov 2005 06:42:43 GMT:
>Is it known now? According to <http://www.mersenne.org/>,
>only 42 are known so far: 2^25,964,951-1 is the 42nd known Mersenne prime
>(but not necessarily the 42nd Mersenne prime, because there might be
>some smaller ones that haven't been discovered yet). Is that web site
>out of date?
All the sites I found agree that the 42nd, discovered in Feb 2005, was the
most recent. Was the OP's instructor trying to get students to discover the
43rd (which may well break the 10-million-digit barrier and win a $100,000
EFF prize), and if so was he planning on keeping the prize for himself?
>If you don't know which (if any) number is the 43rd Mersenne prime,
>then why bother computing the first 25 digits of some particular
>2^x - 1 that might be that prime? You could just write down
>any 25 digits, and that might happen to be the first 25 digits
>of the 43rd Mersenne prime, if there is one.
>In fact, in the absence of any known bound on how big that 43rd
>Mersenne prime could be, its first 25 digits could be any string
>of 25 digits not starting with 0.
If you look at the sequence of MP exponents on a log basis, they appear
to be uniformly (but randomly) distributed.
digits in n number of MPs
----------- -------------
1 4
2 6
3 4
4 8
5 6
6 5
7 5
8 4 so far
Realistically you'd expect the 43rd to have an exponent of 8 digits or
maybe 9, which leaves a relatively small subset of valid 25-digit prefixes.
--Keith Lewis klewis {at} mitre.org
The above may not (yet) represent the opinions of my employer.
.
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