If one defines an elementary number to be a real root or the real part of a complex root of any finite system of elementary functions with integer coefficients, has anyone conjectured in print that every irrational elementary number is a normal number (a real number whose infinite sequence of digits in every base is distributed uniformly)? Are any counterexamples known?

Don N. Page
Professor of Physics
University of Alberta

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    $\begingroup$ Do you have a precise definition of "elementary function" in mind? $\endgroup$ – François Brunault Jan 7 '13 at 20:13
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    $\begingroup$ @Michael: Maybe it is not too hard to cook up an "elementary" definition of a non-normal irrational (whatever the OP means by "elementary", which he needs to clarify.) $\endgroup$ – Sidney Raffer Jan 7 '13 at 20:54
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    $\begingroup$ Are floors allowed in the equation? If so, you can likely construct a non-normal number with techniques similar to how the logicians prove first-order statements quantified on natural numbers with just addition and multiplication can encode anything. If not, I still think it's likely you can do something similar. $\endgroup$ – Zsbán Ambrus Jan 7 '13 at 20:55
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    $\begingroup$ The known conjecture is that algebraic irrational numbers are normal in all bases. So switching "elementary function" to "polynomial" we get this conjecture. Of course no proof is in sight. $\endgroup$ – Gerald Edgar Jan 7 '13 at 21:37
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    $\begingroup$ If one uses instead normality in the continued fraction sense (the sequence of coefficients has exactly the correct statistics), then the corresponding conjecture is false (e for example). This kind of normality was studied by Keane and Smorodinsky $\endgroup$ – Anthony Quas Jan 7 '13 at 21:44

Perhaps connected to general conjectures published here ... "On the Random Character of Fundamental Constant Expansions", David H. Bailey and Richard E. Crandall
Experiment. Math. Volume 10, Issue 2 (2001), 175-190.


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