Normal Numbers and Pseudorandom Generators

  • David H. BaileyEmail author
  • Jonathan M. Borwein
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 50)


For an integer b ≥ 2 a real number α is b -normal if, for all m > 0, every m-long string of digits in the base-b expansion of α appears, in the limit, with frequency b m . Although almost all reals in [0, 1] are b-normal for every b, it has been rather difficult to exhibit explicit examples. No results whatsoever are known, one way or the other, for the class of “natural” mathematical constants, such as \(\pi,\,e,\,\sqrt{2}\) and log2. In this paper, we summarize some previous normality results for a certain class of explicit reals and then show that a specific member of this class, while provably 2-normal, is provably not 6-normal. We then show that a practical and reasonably effective pseudorandom number generator can be defined based on the binary digits of this constant and conclude by sketching out some directions for further research.

Key words

Normal numbers Stoneham numbers Pseudorandom number generators 

Mathematics Subject Classifications (2010)

11A63 11K16 11K45 



The first author is supported in part by the Director, Office of Computational and Technology Research, Division of Mathematical, Information, and Computational Sciences of the US Department of Energy, under contract number DE-AC02-05CH11231.


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Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

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

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