[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.

More information about the SeqFan mailing list