On 13 Jul 99, at 13:41, Todd Sauke wrote, in reply to my
previous message
> Brian Beesley wrote:
>
> >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.
> >
>
> This is not true. Actually, it is the fact that smaller primes are more
> likely to give Mersennes that theoretically should result in a "Benford's
> Law" type behavior of the second leading bit. It is in some sense an
> "accident" that Mersenne exponents SHOULD follow Benford's Law (at least
> the second bit generalization of it), and an irony that, due to small
> number statistics, they actually DON'T! (68% zeroes instead of predicted
> 58% or whatever)
The effect I was predicting was small. Experiments indicate it should
be hard to detect, even with the small numbers of Mersenne primes
known. The relevant fact is that we have reasonable coverage of
exponents up to approx. 5 million, not that we know less than 40
Mersenne primes.
> Benford's Law comes about because of power law scaling of
> some numbers. Many of the referenced web links emphasized that Benford's
> law is NOT for "regular" numbers, but ONLY for numbers expressing amounts
> in some (human selected) units, and that it is some property of "power law
> scaling" and/or "logarithmic invariance" of arbitrary choice of units to
> express AMOUNTS (NOT numbers) of things that result in Benford's law.
>
There is an article on this phenomenon in the current on-line issue
of "New Scientist" (UK), the URL is
http://www.newscientist.com/ns/19990710/thepowerof.html
Regards
Brian Beesley
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