Normality of $\pi$ (open)

Fix an integer base $b \ge 2$. A real number $x$ is normal in base $b$ if for every $k \ge 1$ each of the $b^k$ blocks of $k$ digits occurs in the base-$b$ expansion of $x$ with asymptotic frequency $b^{-k}$; it is absolutely normal if it is normal in every base $b \ge 2$.

Question. Is $\pi$ normal in base $10$? In any base?

Status: open. It is not known whether $\pi$ is normal in a single base. Much less is known than that: it is not even known whether every one of the digits $0, \dots, 9$ occurs infinitely often in the decimal expansion of $\pi$. The same questions are open for $e$, for $\sqrt{2}$, and for $\ln 2$. No naturally occurring constant has ever been proved normal.

Not proved in this library, and not provable anywhere at present. Nothing here depends on any digit statistic of $\pi$.

What is known, and what would settle it. Borel (1909) proved that almost every real number is absolutely normal, so normality is the typical behaviour and the exceptions form a null set; stating that theorem correctly needs the measure notions of the deferred measure and integration track. Explicit normal numbers are easy to write down once one stops asking for a familiar constant: Champernowne's constant $0.123456789101112\ldots$ is normal in base $10$, and Sierpinski and Turing gave constructions of absolutely normal numbers. Computations have checked hundreds of trillions of decimal digits of $\pi$ against the usual statistical tests, which they pass; the published record stood at $314$ trillion digits in November 2025, and the digit statistics of such runs are reported routinely. Passing a statistical test is evidence and not a proof, and no amount of computation can settle an asymptotic frequency. The most concrete programme is Bailey and Crandall's: the Bailey-Borwein-Plouffe formula for $\pi$ reduces base-$2$ normality of $\pi$ to a uniform-distribution statement about a specific chaotic iteration, their "Hypothesis A", which would settle the base-$2$ case if proved. Hypothesis A is open.

Why it matters here. This library defines $\pi$ analytically and will prove sharp facts about it, and a reader is entitled to ask what is not known. The answer is instructive: a constant can be pinned down exactly by a convergent series, be computable to arbitrary precision, and still resist the most basic question about its digits. Normality is also where analysis stops being able to help and measure-theoretic and number-theoretic tools take over, which is why it sits in this category rather than on a page.