32 problems

An irrational algebraic number has a base-$b$ expansion that is normal for every integer base $b\geq 2$. Borel's normality conjecture. Every irrational algebraic number is normal.…

Let $\mathcal{F}$ denote the concatenated Fibonacci constant in base $10$. A real number is normal in base $10$ if every block of $k$ digits occurs with limiting frequency…

Let … be a BBP-type constant, where $s$, $b$, and $l$ are integers denoting the degree, base, and length, respectively, and the $a_j$ are integers. A real number is normal to base…

Normality conjecture. Among the deterministic numbers in base $2$, normality in every base multiplicatively independent of $2$ is a generic property.

Let … be the real number whose decimal expansion is obtained by listing all prime numbers in increasing order. A number is normal in base $s$ if every digit…

A real number is absolutely normal if it is normal in every integer base $b\geq2$. Absolute-normality conjecture. All irrational algebraic numbers and most transcendental constants…

Let $\alpha\in\mathbb{R}$ be an irrational algebraic number. A real number is normal if its digit frequencies are uniform in every integer base. Normality conjecture for irrational…

Let $e$ denote Euler's number, and let $p$ be a non-constant integer-valued polynomial. A real number is normal if its digit frequencies are uniform in every integer base. Euler po…

Let $p$) be a prime and let $\mathbb{Z}_p$ denote the ring of $p$-adic integers. A $p$-adic integer is normal in base $p$ when its base-$p$ digit expansion has every finite block w…

Normality conjecture for irrational algebraic numbers. Every irrational algebraic number is absolutely normal.

A real number has a $b$-ary expansion, and it is normal if every word of length $k$ occurs with frequency $1/b^k$. The normality conjecture. Every irrational algebraic number is no…

Kechris's conjecture. The set of absolutely normal numbers is $\boldsymbol{\Pi}^0_3(\mathbb{R})$-complete.

Low-discrepancy conjecture. Poisson generic numbers in base $b$ cannot have very low discrepancy in their initial segments; in particular, infinite de Bruijn sequences in bases…

Furstenberg's normality conjecture. For every $\mu\in\mathcal{P}_{p,c}({\mathbb T})$, for $\mu$-almost every $x\in{\mathbb T}$ and every $a\in{\mathbb Z}\setminus\{0\}$,

Let $b$ and $m$ be as in Theorem 2, and consider normal numbers constructed via $(b^m,b^m)$-nested perfect necklaces. Generalization conjecture. Theorem 2 should extend to all norm…

Let $L$ be a number field and let $\alpha\in L$ be irrational. Let $\mathfrak{a}$ be an unramified ideal of degree one that is prime to the denominator of the principal ideal…

Let $\alpha$ be an irrational algebraic number and let $n>1$ be an integer. An $n$-adic expansion of $\alpha$ is a representation $\sum_{l=0}^{\infty}a_ln^l$ with $0\le a_l<n$ sati…

Let $\{\varphi_i(x)=\lambda_i x+t_i\}_{i\in \mathcal{I}}$ be an iterated function system of contracting similarities, and let $b\geq 2$ be an integer. Suppose that … for some…

Normality conjecture for $\pi$. The constant $\pi$, and other natural transcendental constants, should be normal in every base $b\geq 2$.

Let $\mathbf J$ be the jimm involution and let $X$ denote a jimm-transformed number in the experimental setting. Rational-scaling law conjecture. $q\mathbf J(X)$ obeys a certain la…

Let $\mathcal N$ denote the set of normal real numbers in the sense of obeying the Gauss–Kuzmin statistics, and let $\mathbf J$ be the jimm involution. Normality conjecture. If…

Applicability conjecture. It is not clear whether this approach can be used to investigate the normality of well-known irrational numbers conjectured to be normal, such as … and ……

Let $b\geq 2$ be an integer, and write the base-$b$ expansion of $\sqrt{2}$ as an infinite digit sequence. For each finite string of digits of length $k$, consider its limiting fre…

Let $b\geq 2$ be an integer base, and let $(\alpha_1,\ldots,\alpha_n)$ be an $n$-tuple of real numbers. Writing their base-$b$ expansions in aligned columns, call the tuple normal…

Let $T$ and $S$ be number-theoretic transformations on the same space $\mega$, with normal-equivalence meaning that a point $x$ is $T$-normal if and only if it is $S$-normal. Schwe…