51 problems

Nivat's conjecture. Then $c$ is $1$-periodic.

Periodic orbit-closure conjecture. The orbit closure

For positive integers $n$ and nonnegative integers $d$, let $G_n(d)$ be the graph whose vertices are the monomials of degree $d$ in $n$ variables, with two monomials adjacent exact…

Let $f$ be a transcendental entire function and let $k$ be a positive integer. Yang's periodicity conjecture. If $k\geq 1$ and $ff^{(k)}$ is periodic, then $f$ must be periodic. Th…

Let $(B(t))_{t\in\mathbb{Z}}$ be the exchange-matrix sequence and let $(Y(t))_{t\in\mathbb{Z}}$ and $(y(t))_{t\in\mathbb{Z}}$ denote, respectively, the corresponding quantum and no…

Period Conjecture. For each integer $j=1,\dots,d$, there is a positive constant $c_j(r,R)$ such that, if for some $t_0(X)$

Let $f$ be a transcendental entire function and let $k$ be a positive integer. For $k\geq 2$, suppose that either $ff^{(k)}$ is periodic or … for some entire function $h$. Yang's p…

Let $\opL$ be an irreducible object with associated set $\Lambda_\opL$, preorder $\preceq$, period $N_\alpha$, and invariant $\varsigma_\alpha$ for each $\alpha\in\Lambda_\opL$. Ma…

Main horizontal-period conjecture. Under these assumptions,

Lengyel's conjecture. (a) If $b$ is odd and at least $3$, then $L(b;b,0;1,1)$ has period $(2b(b+1),2b)$. (b) For $b\geq 2$, $L(b;1,0;1,1)$ has period $(4b,2b)$.

Let $M = \prod_{p|M} p^{a_p}$ and $N = K \cdot \prod_{p|M} p^{b_p}$ be integers with $a_p, b_p \ge 1$ and $\gcd(K,M) = 1$. For each prime factor $p|M$, define … The sequence $b(n)$…

Let $p$ be prime and let $M$ be a $C_p$-Mackey functor. A projective resolution of $M$ is eventually $4$-periodic if its differentials repeat with period $4$ from some point onward…

Let $G$ be a Garside group equipped with a Garside structure, and let $SC(x^n)$ denote the sliding circuits set of $x^n$. Uniform periodicity conjecture. There exists a finite set…

Periodicity conjecture. These $h_0$-submodules have the same $h_0$-coexponent for all $i$.

Let $[a_0,a_1,\ldots]$ be a continued fraction with $a_n\in\mathbb{Q}$ for all $n$. Suppose it converges simultaneously in $\mathbb{R}$ and $\mathbb{Q}_p$ to a real quadratic irrat…

Let $[a_0,a_1,\ldots]$ be the Browkin I, Browkin II or Algorithm MR expansion of a $p$-adic quadratic irrational $\alpha^{(p)}$ that can be embedded in the real numbers. Let…

Let $S/(h,v^4_{(1,0)})$ denote the indicated $C_2$-equivariant complex, let $a$ be the Euler-class element, and let $\widetilde{\bar{\kappa}}$ denote the specified equivariant lift…

Non-eventual-periodicity conjecture. Not all finite two-dimensional rulesets are eventually periodic.

Three-move segmentation conjecture. If $|S|=3$, there is an outcome segmentation.

Let $p>3$ be a prime, let $n_1,n_2>1$, and let $E_n(1,x)$ denote the reversed Dickson polynomial of the second kind over $\mathbb{Z}_p$. Reversed Dickson second-kind period conject…

Let $m$ range over the positive integers, and consider the Jacobi–Perron algorithm (JPA) expansion of … A positive integer $m$ is called eventually periodic when this expansion is…

Let $p$ be a prime with $p\equiv1\pmod4$, let $e\ge1$, and let $(d(n))_{n\ge1}$ be the sequence defined in the paper. The fourth conjecture. (1) The sequence $d(n)$ modulo $p^e$ is…

For each integer $k\geq 4$, let $RT'(k)$ denote the threshold exponent associated with the existence of infinite pseudoperiodic words in the paper's notation. Critical exponent con…

Let $(a,b)$ be a pair of positive integers with $1\leq a<b$. An infinite binary word has pseudoperiod $(a,b)$ if each position is compatible with a repetition having periods $a$ an…

A segment of $n$ vertices starting with a black vertex is denoted by $S_n$, and a segment starting with a white vertex by $S_{-n}$. Let $Ls(S_n)$ and $Rs(S_n)$ denote the left and…