Open problem, Arithmetic and number theory, Probability and statistics, posed 1909

Normality of π

Open

Is $\pi$ a normal number in base $10$ (or in every integer base $b \ge 2$ ) — that is, in the base- $b$ expansion of $\pi$ , does every finite block of $k$ digits occur with limiting asymptotic frequency $b^{-k}$ ?

Research frontier as of 2026

As of 2026, the normality of $\pi$ remains completely open in every base $b \ge 2$ . Computations of $\pi$ past $10^{14}$ decimal digits reveal no statistically significant deviation from the uniform frequency $10^{-k}$ for short digit blocks, and the Bailey–Crandall (2001) framework reduces base- $2$ normality of BBP-type constants to equidistribution of perturbed doubling maps $x_n = (2 x_{n-1} + r_n) \bmod 1$ . While Bailey and Crandall succeeded in proving normality for explicit Stoneham-type constants such as $\alpha_{2,3} = \sum_{k=1}^\infty \frac{1}{3^k 2^{3^k}}$ where the denominators grow lacunarily, the linear denominator polynomial $8k+j$ in the BBP series for $\pi$ lacks the lacunarity needed to bound Weyl exponential sums unconditionally.

Best known results

Borel (1909): the set of real numbers that fail to be normal in every integer base $b \ge 2$ has Lebesgue measure $0$ .
Bailey and Crandall (2001): $\pi$ is normal in base $2$ and base $16$ if the sequence $x_n = (16 x_{n-1} + \frac{120n^2 - 89n + 16}{512n^4 - 1024n^3 + 712n^2 - 206n + 21}) \bmod 1$ with $x_0 = 0$ is equidistributed in $[0, 1)$ .

Tools and where they stop

Tool	Achieved	Where it stops
BBP digit-extraction formulas and Weyl equidistribution (Bailey–Crandall)	Translates base- $16$ normality of $\pi$ into equidistribution of an explicit 1D recurrence and proves normality for lacunary BBP-like constants	For $\pi$ , the perturbation term $r_n = O(n^{-2})$ is non-lacunary, so iterated powers $16^{n-k}$ amplify errors before Weyl sums can be averaged
Multi-trillion-digit computation and statistical randomness testing	Confirms empirical equidistribution of digits and short blocks across more than $10^{14}$ decimal and hexadecimal digits of $\pi$	Normality is an asymptotic property as $N \to \infty$ ; any finite prefix of digits is compatible with both normality and complete failure of normality

Open questions

Does every digit $0, 1, \dots, 9$ occur infinitely often in the decimal expansion of $\pi$ ?
Does the Bailey–Crandall equidistribution hypothesis hold for the BBP recurrence associated with $\pi$ ?

References

Émile Borel (1909). Les probabilités dénombrables et leurs applications arithmétiques · DOI:10.1007/BF03019651
David H. Bailey, Peter B. Borwein, Simon Plouffe (1997). On the rapid computation of various polylogarithmic constants · DOI:10.1090/S0025-5718-97-00856-9
David H. Bailey, Richard E. Crandall (2001). On the random character of fundamental constant expansions · DOI:10.1080/10586458.2001.10504441
Yann Bugeaud (2012). Distribution Modulo One and Diophantine Approximation · DOI:10.1017/CBO9781139017732