[seqfan] Re: Pi Day Question

[Jeffrey Shallit]

> I believe that the number
> 0.12345678910111213141516171819202122232425262727293-031.. . .
> was proved to be normal many, many years ago. I heard a rumor fifty-two
> years
> ago via my number theory teacher, the late Alfred T. Brauer (at Univ. of
> N.C.)
> that perhaps a Russian mathematician had proved that the number
> 0.23571113171923293137 . . . formed by joxtaposing the digits of the prime
> number sequence, is normal; however I do not recall finding a reference.
> Can any Sequence Fanatic answer this and can anyone cite other specific
> numbers
> which are normal?

Champernowne's number .12345678910... is normal to base 10, but not known
to be normal in other bases.  See Champernowne, J. London Math. Soc.
(1933), 254-260.

The number .23571113171923.... was proved normal in base 10
by Copeland and Erdos, Bull Amer. math. Soc. 52 (1946), 857-860, but it is
not known to be normal in other bases.

Jeffrey Shallit