24 problems

Power-weight duality conjecture. The power weights $w^{(\ell)}$, $\ell \geq 2$, do not respect duality over any $\mathbb{Z}/m\mathbb{Z}$ with $m \geq 4$.

Let $R$ be a finite non-commutative CC-ring, meaning that all centralizers of non-central elements of $R$ are commutative, and let $\Gamma_R$ be its commuting graph. Let MSN-integr…

Let $R$ be a finite CC-ring, meaning that the centralizer of every non-central element of $R$ is commutative, and let $\Gamma_R$ be its commuting graph. A graph is CN-hyperenergeti…

Let $p_n$ denote the $n$th prime, and define … Let $\mathcal{C}_{\mathcal{A}}$ denote the subring of algebraic elements of $\mathcal{A}$. Transcendence conjecture. … The paper prov…

Let $n$ be a natural number, and let $G(\mathbb{Z}_n)$ be the unit graph. Write $U(\mathbb{Z}_n)$ for its set of units, $N_U(\mathbb{Z}_n)$ for its set of non-units, and…

Let $p$ be a prime, and let $g_s(p)$ denote the least stationary primitive root in $\mathbb{Z}/p\mathbb{Z}$. Let $\varepsilon>0$ be a small number. Least stationary primitive root…

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 $\mathfrak{g}$ be a nilpotent $\mathbb{Z}$-Lie algebra finitely generated as an abelian group. For a finite truncated valuation ring $R$ with maximal ideal $\mathfrak{m}$, writ…

Let $f$ be a polynomial in $n$ variables, with $s$ and $d$ as in the preceding setup, and define … Here $N\geq 1$ and $\psi:\mathbb Z/N\mathbb Z\to\mathbb C^\times$ is a group mono…

Let $N,n\in\mathbb{N}$, and let $S\subseteq (\mathbb{Z}/N\mathbb{Z})^n$ be a Kakeya set, meaning that $S$ contains a line in every direction in…

Let $R=\mathbb{Z}_{n}$, where … with primes $p_{1}<p_{2}<p_{3}<\cdots<p_{k}$, including $2$. Write $\psi(n)$ for the quantity appearing in the conjecture. Maximal-index conjecture.…

Middle-weight conjecture. Then $w_2=n$.

Soit $A$ un $p$-anneau fini, c'est-à-dire un anneau fini dont la caractéristique est une puissance de $p$, et soit $K$ un corps tel que $\mathrm{car}(K)\neq p$. Pour un foncteur…

Let $n>1$ be an integer, and let $R=\mathbb{Z}_n$ be the ring of integers modulo $n$. The modular Erdős–Burgess constant conjecture. … This conjecture gives an exact formula for th…

Let $\gamma(t)=(t,t^2,\ldots,t^n)$ and define $\Phi(\vec t)=\sum_{i=1}^n\gamma(t_i)$. For $\vec t\in[\mathbb Z/N\mathbb Z]^n$, let $\mathbf N(\vec t;N)$ be the number of…

Let $n\geq 2$, let $1\leq r,s\leq\infty$, and write $r'$ for the Hölder conjugate of $r$. Let the discrete restriction estimate refer to the estimate defined earlier in the source.…

Let $n\geq 2$, let $N$ be an odd positive integer, and let $\mathcal{E}$ denote the extension operator for the paraboloid over $\mathbb Z/N\mathbb Z$, acting on functions $H$ on…

Class 1 conjecture. The relative non-commuting graph $\Gamma_{S,R}$ is of class $1$.

Finite-ring product-of-difference-sets conjecture. There exist $N$ and $k$, depending only on $\alpha$, such that for every finite commutative ring $R$ with $|R|\ge N$ and every se…

Fortuny–Grau–Oller-Marcén–Rúa conjecture. All such power sums vanish unless all of the following conditions hold: $d=2$; $|R|\equiv 2\pmod 4$ and $1<k\equiv 0,\pm1\pmod 6$; and the…

Let $R$ be a finite commutative unital ring, and let $|R|$ denote its cardinality. Generalized Giuga conjecture. … if and only if $R$ is a field. This generalizes the classical Giu…

Let $d>1$ and let $R$ be a finite commutative ring. For $k\geq 1$, write $S_k(\mathbb{M}_d(R))=\sum_{A\in\mathbb{M}_d(R)}A^k$. The unique element $e\in R\setminus\\{0\\}$ such that…

Non-commutative power-sum conjecture. The theorem for finite commutative rings remains true for non-commutative $R$: $S_k^d(R)=0$ unless $d=2$,…

Let $R$ be an arbitrary ring with Jacobson radical $J$, and suppose that … where each $F_i$ is a finite field and $n_i\geq 2$ for all $i$. For a finite ring $A$, let $\tau(A)$ deno…