# Normal Numbers in different bases

If a number is normal in a base $b$, it is absolutely normal. Is this an open question or is there a counter example?

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## 1 Answer

If $r$ and $s$ are multiplicatively independent integers, then it is known that there are (lots of) numbers that are normal to base $r$ but not to base $s$. I think Wolfgang Schmidt may have proved this, and there is also work by Andy Pollington.

EDIT: a reference is Wolfgang M. Schmidt, Über die Normalität von Zahlen zu verschiedenen Basen, Acta Arith. 7 1961/1962 299–309, MR0140482 (25 #3902). This builds on earlier work of Cassels.

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Here's a link to a freely accessible copy: matwbn.icm.edu.pl/ksiazki/aa/aa7/aa7311.pdf –  joriki Oct 22 '12 at 22:49