30 problems

Periodic orbit-closure conjecture. The orbit closure

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

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…

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 $S/(h,v^4_{(1,0)})$ 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 $p$ 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…

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…

Twin conjecture. There exists $n_0=n_0(p)$ such that for all $n\geqslant n_0$ and each group $G$ at depth $n-c$ in $\mathcal B_p(n+d)$, there exists a twin $H$ at depth $n-c-d$ in…

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 $KAYLES_n$ denote the Kayles graph game on a path of edge-length $n$, and let $\boxplus$ denote the selective compound operation used in the paper; write $*1$ for the one-point…

Let $STARKAYLES_{k,n}$ be the graph game in which the underlying graph is obtained by starting with a star graph on $k$ vertices and extending one branch to a path of edge-length…

An octal game is a game played with tokens divided into heaps, where a move either removes some or all tokens in one heap, or removes some but not all tokens from a heap and divide…

Finite-family Hermite matrix conjecture. There is a family of multidimensional continued fractions spanned by finitely many multidimensional continued fraction algorithms such that

Let $b\geq 2$ be an integer, and let $\mathfrak{S}_b(n)$ denote the last non-zero digit of $n^n$ in base $b$. Periodicity conjecture. The sequence $\mathfrak{S}_b(n)$ is eventually…

Let $p$ be a prime, let $M$ be a topological space, and let $x\in H^k(M;\mathbb{Z}_p)$ be nonzero. Say that $x$ induces periodicity up to degree $r$ when multiplication by $x$ give…