The Ramanujan Journal

, Volume 29, Issue 1–3, pp 409–422 | Cite as

Nonnormality of Stoneham constants

  • David H. Bailey
  • Jonathan M. BorweinEmail author


This paper examines “Stoneham constants,” namely real numbers of the form \(\alpha_{b,c} = \sum_{n \geq1} 1/(c^{n} b^{c^{n}})\), for coprime integers b≥2 and c≥2. These are of interest because, according to previous studies, α b,c is known to be b-normal, meaning that every m-long string of base-b digits appears in the base-b expansion of the constant with precisely the limiting frequency b m . So, for example, the constant \(\alpha_{2,3} = \sum_{n \geq1} 1/(3^{n} 2^{3^{n}})\) is 2-normal. More recently it was established that α b,c is not bc-normal, so, for example, α 2,3 is provably not 6-normal. In this paper, we extend these findings by showing that α b,c is not B-normal, where B=b p c q r, for integers b and c as above, p,q,r≥1, neither b nor c divide r, and the condition D=c q/p r 1/p /b c−1<1 is satisfied. It is not known whether or not this is a complete catalog of bases to which α b,c is nonnormal. We also show that the sum of two B-nonnormal Stoneham constants as defined above, subject to some restrictions, is B-nonnormal.


Normality of irrational numbers Nonnormality of irrational numbers Stoneham numbers Normality of sums 

Mathematics Subject Classification

11K16 11K31 



The authors wish to express their profound appreciation to Boris Adamczewski who, as referee, made a very incisive report. This provided Lemma 2 and sketched proofs of Theorems 2 and 3. Lemma 2 in particular appears to be a very useful additional tool for research into the normality of mathematical constants.


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Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.Lawrence Berkeley National LaboratoryBerkeleyUSA
  2. 2.Centre for Computer Assisted Research Mathematics and its Applications (CARMA)University of NewcastleCallaghanAustralia

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