[seqfan] Re: Pi Day Question
Jonathan Post
jvospost3 at gmail.com
Sat Mar 14 21:11:49 CET 2009
In mathematics, a normal number is a real number whose digits in every
base show a uniform distribution, with all digits being equally
likely, all pairs of digits equally likely, all triplets of digits
equally likely, etc.
While a general proof can be given that almost all numbers are normal,
this proof is not constructive and only very few concrete numbers have
been shown to be normal. It is for instance widely believed that the
numbers √2, π, and e are normal, but a proof remains elusive.
On Sat, Mar 14, 2009 at 1:07 PM, Leroy Quet <q1qq2qqq3qqqq at yahoo.com> wrote:
>
> [Whoops. I accidently sent this to seqfan at seqfan.com. Resending to correct address now...]
>
> Happy Pi Day, everybody! (At least in the US, where it is still March 14 and where the date is written 3/14.)
>
> A pi question:
>
> Consider the simple continued faction of pi. (The terms of which are sequence A001203.)
>
> The comment at the related sequence A032523 (A032523(n) = the index of the first occurrence of n in A001203) suggests that it is not known for certain that every positive integer occurs in the simple continued fraction of pi.
>
>
> 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?
>
>
> I am not even sure, from what little I have read, that if is known that 1 occurs infinitely often in the continued fraction of pi.
>
> Can anyone enlighten me (and us)?
>
> Thanks,
> Leroy Quet
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