114 problems

An automatic number is a real number generated by an automatic sequence; an $S$-number is a transcendental number with finite, nonzero Diophantine exponent in the relevant classifi…

Sharp upper-bound conjecture. For all $d \geq 2^{2|v|+1}$,

Automaticity conjecture. There is a polynomial $P\in\mathbb{F}_q(t)[x]$ such that

Let $a$ be a multiplicative automatic sequence, and let $g$ be a function on the positive integers. A function is eventually periodic if it agrees with a periodic function for all…

Let $k$ and $l$ be multiplicatively independent positive integers. Let $F\in K[[x]]$ be a power series over a number field $K$ that is both $k$-Mahler and $l$-Mahler. Loxton and Va…

An automatic real number is a real number generated by a finite automaton, and a Liouville number is a real number $\xi$ such that, for every positive integer $n$, there are infini…

Let $\mathbf B=a_0a_1a_2\cdots$ be the infinite binary word obtained from the inflation sequence, and let $I_1(n)$ be the index, using Kimberling's indexing starting at $1$, of the…

Rampersad–Shallit conjecture. The sequence of Motzkin numbers modulo $p$ is uniformly recurrent if and only if $T_n\bmod p$ is never $0$.

The Thue–Morse odd-difference bound conjecture.

Let $t(n)$ denote the Thue–Morse sequence, and let $P(x)$ be a non-negative integer-valued polynomial. Gelfond's conjecture. The values of $t(P(n))$ are equidistributed between $0$…

Becker's conjecture. If $F(z)$ is a $k$-regular power series, then there exists a nonzero $k$-regular rational function $R(z)$ such that $F(z)/R(z)$ satisfies a Mahler-type functio…

Beukers's conjecture. If the representation contains $\alpha$ occurrences of the digit $5$, then

Let $q$ be a positive power of a prime, and let $\textbf{S}$ be an automatic sequence over $\mathbb{F}_q$. Let $W_q(\textbf{S})$ denote the number wall associated with $\textbf{S}$…

Let $q$ be a prime power, let $P(t)\in\mathbb{F}_q[t]$ be irreducible, and let $\Theta(t)\in\mathbb{F}_q(\!(t^{-1})\!)$ be a quadratic irrational. For each $k$, let $\ell_k$ be the…

Let $B_0=00$ and construct the finite words $B_i$ by factoring $B_i$ into maximal blocks of the forms $00$, $0$, and $1$, then replacing these blocks according to $00\mapsto0101$,…

Hansel–Safer conjecture. Assuming the four exponentials conjecture, if $S$ is both $\alpha$- and $\beta$-recognizable, then $S$ is eventually periodic.

Let $A_n$ be the anti-Pell sequence, the anti-recurrence sequence associated with $cmathbf a=(1,2)$. Kimberling's anti-Pell boundedness conjecture. The sequence satisfies … The pap…

Let $cmathbf a=(a_1,cdots,a_k)$ be a positive integral vector of dimension $k>1$, and let $f(cmathbf x)=cmathbf acdotcmathbf x$ be its linear form. Let $A_n$ be the anti-recurrence…

An anti-recurrence sequence is the increasing sequence generated by a positive integral linear form, by selecting the integers represented by that form without repetition. A linear…

Let $a(n)=\sum_{0\leq i\leq n}\binom{n}{i}\binom{n+i}{i}$ be the central Delannoy number, and let $b(n)=\nu_3(a(n))$. Shallit's central Delannoy valuation conjecture. The sequence…

Let $f$ be a multiplicative automatic sequence, and let $g$ be an eventually periodic sequence. Bell–Bruin–Coons' conjecture. For each multiplicative automatic sequence $f$, there…

For each $n\in\mathbb{N}^{\ast}$, let $s_n=(1,\ldots,1,0)$ with $n$ entries equal to $1$. For each $a\in\mathbb{N}$, write its binary representation as $(a_i)_{i\in-\mathbb{N}}$, w…

Let ${\bf{t}}$ be the Tribonacci sequence, fixed point of $0\mapsto 01$, $1\mapsto 02$, $2\mapsto 0$. For each $k\ge 1$ and $n\ge 0$, let $\rho^{k}_{\bf{t}}(n)$ denote its generali…

For an irrational number $\xi$, define the deterministic random walk by … Call $r$ a record if $S_r(2\xi)$ has not occurred among the preceding partial sums, including…

Let $S_n(\xi)=\sum_{j=1}^n(-1)^{\lfloor j\xi\rfloor}$ be a deterministic random walk, and call an integer $n$ a record when $S_n(\xi)$ has not occurred among the preceding partial…