> > Note that in the set of mersenne prime exponents (so far), the leading
> > digit 1 (in decimal), turns up 10 times as opposed to the 4.2 times
> > expected by equal leading digit distribution...
>
> Actually we should expect an excess of smaller leading digits over
> that predicted by "Benford's Law" in this case. A smaller exponent is
> more likely to be prime than a larger exponent, and a smaller prime
> exponent is more likely to give rise a Mersenne prime than a larger
> prime exponent. "Benford's Law" would follow if _every_ exponent
> (prime or composite) was equally likely to give rise to a Mersenne
> prime.
But that's part of the interesting thing... the size of the exponent is
only vaguely associated with the LEADING digit. I disagree that we'd
expect smaller leading digits, at least noticeably many, since it's the
order of magnitude, not just the leading digit that makes the nth mersenne
prime larger or smaller. I mean M20 is some 50 digits longer than M19...
at this distance, I don't see how the leading digit is affected by the
larger likelyhood that smaller exponents would make more likely primes.
But I'm probably wrong. This is what makes this a really interesting
fact, tho, I guess.
> > Yes. Though they were talking about the exponents...
> > Weird, I would have thought that it wouldn't affect powers of
> > two...
>
> Why not? Looks like a perfect model to me!
In some vague attempt to not take the Benford issue off topic, it's
interesting that numbers 2^n (for all Natural numbers n) follows the
pattern VERY closely, but it's also interesting (perhaps moreso), that 2^p
follows the pattern, as do (apparently) the Mersenne primes themselves,
as do (from quick examination) the exponents for the mersenne primes.
Thanks for adding this to the FAQ, btw.
---Chip
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| Chip Lynch | Computer Guru |
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