Steven Whitaker wrote:
> Maybe it's my imagination, but it seems to me that the factors of the
> prime exponent Mersenne numbers start with a 1 more often than a 2 or
> 3 etc. Are they obeying Benford's law too?
> For instance, for the 10 primes from 5003 to 5081, there are 20 known
> factors. 10 of them start with a 1.
Something similar. The Benford's law distribution works because we 'expect'
natural, boundless, data to have the decimal part of the logarithm "fairly
uniformly" distributed, and a quick look at a slide rule (younger readers,
ask your Dad!) has 30.103% of its length with initial digit 1.
By Merten's theorem, the probability *any* large number N has no factor
smaller than X is C/log X, C is exp(-gamma) if I remember rightly.
(strictly, X has to be much bigger than 1, and much smaller than sqrt(N) for
this to make any sense). By that sort of estimate, suppose N has a factor F
between 10^k and 10^(k+1).
Then the probability that F begins with a 1 is something like
[1/k-1/(k+0.30103)]/[1/k-1/(k+1)]
which tends to log10(2) as k tends to infinity - so the distribution does
approach Benford's law. In fact, if you plot the distribution above, it's
more generous to 1's for smaller factors. It seems a skewed distribution,
but remember it's based on our insistence of observing the world base 10,
and believing that 2-1 is "just as important" as (M38)-(M38-1). The same
observation is repeatable in any base - of course, at the most ridiculous,
ALL factors begin with a 1 when written in base 2.
Chris Nash
Lexington KY
UNITED STATES
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Still a co-discoverer of the largest known *non*-Mersenne prime
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