A Study on the Randomness of the Digits of Π

Abstract

We apply a newly-developed computational method, Geometric Random Inner Products (GRIP), to quantify the randomness of number sequences obtained from the decimal digits of π. Several members from the GRIP family of tests are used, and the results from π are compared to those calculated from other random number generators. These include a recent hardware generator based on an actual physical process, turbulent electroconvection. We find that the decimal digits of π are in fact good candidates for random number generators and can be used for practical scientific and engineering computations.

An experimental investigation of the normality of irrational algebraic numbers

“…If Borel's conjecture is proved, it could conceivably be devoid of practical significance: Even if the distribution of the set of digits in successively larger prefixes of the expansions of a number converges towards equidistribution, the convergence could be so slow that for all prefixes that can realistically be computed, the distribution is skewed and far from equidistribution (experimental investigations on the digit distribution of π suggest that if π is normal, then the prefixes of its expansion to certain bases do converge rapidly [7,45]). …”

“…In addition, for specific algebraic numbers, very large prefixes have been computed, and for specific transcendental numbers such as e and π, large prefixes of their expansions have been subjected to statistical analysis [44,47,26,7,45].…”

Geometric Random Inner Product Test and Randomness of Π

“…By using this approach, we have calculated the parametric values of means and variances presented in Tables 1 and 2 4 we infer that, for example, in their Table 1, the sample standard error of NSW is 0.00206, whereas it is 0.00159 for most entries. In Table 2 for n = 3, the standard error is about 0.00248, but NWS has a value of almost double: 0.00471.…”

“…Our experiments on large moduli showed that in GRIP test, higher resolution of generated numbers can even hide the deficiency of generators having extremely bad lattice structure such as X n = 7X n−1 (Mod 2 31 − 1). Therefore for Tables 5 and 6 of Tu and Fischbach, 4 we examined the following set of generators and presented the test results in Tables 2-5: (1) LCG1: The multiplicative random number generator X n = 16 807X n−1 (Mod 2 31 − 1). (2) F55a: The lagged Fibonacci generator using X n = (X n−55 + X n−24 ) (Mod 2 31 ).…”

“…In their recent papers, 1,3 Tu and Fischbach presented a newly developed test and called it Geometric Random Inner Product (GRIP). In a previous paper, 4 they investigate the randomness of digits of π by using this test. Although this work supports the opinion of previous authors 5,8 for the appropriateness of π as a random number generating source, finds it inferior to some other generators.…”