[seqfan] Re: Pi Day Question
Hans Havermann
pxp at rogers.com
Sun Mar 15 00:22:52 CET 2009
Leroy Quet:
> Can it be said, however, that either: it is known that each positive
> integer that does occur occurs infinitely often; OR it is known that
> at least some integers occur finitely often?
Just to steer this slightly away from a discussion of "normal" numbers
which has always struck me as referring to the frequency of digits in
a given base (not necessarily to a simple continued fraction
expansion), in 1935 Aleksandr Khinchin showed that, for almost all
irrationals, the frequency of occurrence of a given number 'n' in its
(infinite) simple continued fraction expansion is (Log[(n + 1)^2] -
Log[(n + 1)^2 - 1])/Log[2] ("Log" here being the natural logarithm,
i.e., to the base 'e').
My take on this has always been that every positive integer will occur
infinitely often.
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