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An empirical approach to the normality of \(\pi\). (English) Zbl 1281.11077

Summary: Using the results of several extremely large recent computations by A. J. Yee and S. Kondo [10 trillion digits of pi: a case study of summing hypergeometric series to high precision on Multicore Systems http://hdl.handle.net/2142/28348], we tested positively the normality of a prefix of roughly four trillion hexadecimal digits of \(\pi\). This result was used by a Poisson process model of normality of \(\pi\): in this model, it is extraordinarily unlikely that \(\pi\) is not asymptotically normal base 16, given the normality of its initial segment.

MSC:

11K16 Normal numbers, radix expansions, Pisot numbers, Salem numbers, good lattice points, etc.
03D32 Algorithmic randomness and dimension
11A63 Radix representation; digital problems

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