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) 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.
As has been dealt with in many many recent posts regarding the density of
Mersenne exponents, there is an expected (and observed) uniform density of
Mersennes in LOG space, ie. equal numbers of Mersennes per factor of two in
candidate space of about 1.5 or 1.48 or whatever. It is this logarithmic
scaling or invariance that theoretically SHOULD result in Benford Law type
behavior. It is ONLY this decreasing likelihood of primes to generate
Mersennes that should cause Benford law behavior, not a reason for
deviation from it. In fact, if the likelihood of numbers generating
Mersennes followed some other law, say decreasing faster than
logarithmically, then Benford's law might not apply, or only approximately.
This thread has been very interesting in it's own right mathematically, but
it's only "accidental" in some sense that Mersenne exponents should follow
the law, and as I said before, ironic that it doesn't! The fact that we
KNOW (or believe) the logarithmic scaling of Mersennes allows us to bypass
all of the heuristic arguments about logarithmic scaling of human selected
units for expressing quantities, and to directly DERIVE the law for
Mersennes, as opposed to heuristically generating it for the other cases.
Todd Sauke
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