[seqfan] Re: Pi Day Question

[Hans Havermann]

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.