Normal Numbers and Pseudorandom Generators∗

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 π, e, √ 2 and log 2. 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. ∗Submitted to Heinz Bauschke, ed., Proceedings of the Workshop on Computational and Analytical Mathematics in Honour of Jonathan Borwein’s 60th Birthday, Springer, 2011. †Lawrence Berkeley National Laboratory, Berkeley, CA 94720, [email protected]. Supported in part by the Director, Office of Computational and Technology Research, Division of Mathematical, Information, and Computational Sciences of the U.S. Department of Energy, under contract number DE-AC02-05CH11231. ‡Laureate Professor and Director Centre for Computer Assisted Research Mathematics and its Applications (CARMA), University of Newcastle, Callaghan, NSW 2308, Australia. Distinguished Professor, King Abdulaziz University, Jeddah 80200, Saudi Arabia. Email: [email protected].