How do you define normal to factorial base? - Scott On Wed, Aug 12, 2015 at 8:32 PM, Dan Asimov <dasimov@earthlink.net> wrote:
Do you mean normal to a specific base? Given a base b, it's easy to construct uncountably many numbers normal to that base by just making sure that every possible string of N digits occurs, in the limit, the fraction 1/b^N of the time.
Normal to every base seems much harder.
I'm also interested in normal to factorial base, which seems more natural than normal to base b or even to all bases.
—Dan
On Aug 12, 2015, at 8:15 PM, Keith F. Lynch <kfl@KeithLynch.net> wrote:
[Proving transcendence is} nothing compared to normality. Again "nearly all" real numbers are normal, but it's not easy to prove that any specific number is normal. Pi, for instance, was proven irrational in the 18th century and transcendental in the 19th, but the 20th didn't accomplish anything with it except increase the number of known digits from a few hundred to a few hundred billion. And the 21st has done nothing except increase the number of known digits by another factor of 100 or so. Maybe the 22nd century will prove (or disprove) that it's normal. Or maybe pi's normality is literally unknowable.
Are any specific numbers known to be normal? That's not clear to me. Wikipedia says, without references:
.... This theorem established the existence of normal numbers. In 1917, Wacl/aw Sierpinski showed that it is possible to specify a particular such number. Becher and Figueira proved in 2002 that there is a computable absolutely normal number, however no digits of their number are known.
What is the specification of Sierpinski's number? And what is the algorithm for computing Becher/Figueira's number, and why have no digits been computed? Thanks.
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