102 problems

Let $T\subseteq \mathbb{Z}_v\setminus\{0\}$ satisfy … An ordering of the elements of $T$ is simple when its partial sums are all distinct and non-zero. Alspach's conjecture. Every…

Let $k$ be an infinite base field of characteristic $p$, where $p$ is prime, and let $n\geqslant 1$ be an integer. Write $\mathbb Z/p^n\mathbb Z$ for the cyclic group of order…

A strong starter in the cyclic group $\mathbb{Z}_n$ for odd $n$ is a partition of $\{1,2,\dots,n-1\}$ into pairs $\{a_i,b_i\}$ such that the pair sums $a_i+b_i$ are distinct and no…

Erdős–Lemke conjecture. Every such sequence contains a zero-sum subsequence whose sum is at most $\operatorname{lcm}(n,a_1,\ldots,a_n)$.

Zhang–Ge conjecture. Let $k_1, \,k_2$ be integers with $1 \le k_1 < k_2$ and $k_1 + k_2\ge 4$. Then there does not exist any purely singular perfect $B-k_1, \,k_2$ set except for…

Archdeacon–Dinitz–Mattern–Stinson conjecture. For any cyclic group $\mathbb{Z}_n$ and any subset $A\subseteq\mathbb{Z}_n\setminus\{0\}$, there is an ordering of the elements of $A$…

Let $k$ be fixed, and let $A$ be a $k$-element subset of the nonzero elements of ${\mathbb Z}_n$. Fixed-size sequencing problem. Determine whether there exists a constant $N$ such…

Cyclic homology recursion. For every positive integer $n$, one has $H_i(({\mathcal C}^{C_{2n}},D))=0$ when $i$ is odd, so that $A_{2n}(q)$ is a polynomial in $q^2$, and

Richman's conjecture. The elements $x_i$, $N_i$, $u_{ij}$, and $\operatorname{tr}(A)$ for $0\leq a_i\leq p-1$ generate the invariant ring ${\bf F}[mV_2]^{C_p}$. Richman proved the…

Cyclic Cayley graph optimality conjecture. These graphs are optimal Abelian Cayley graphs of the specified type for every $k\geq 3$, not only for $3\leq k\leq 10$.

Let $p$ be the characteristic of ${\bf F}$, let $V_n$ denote the relevant indecomposable ${\bf F}G$-module of dimension $n$, and let ${\bf F}[V_n]^{\flat}_d$ be the degree-$d$ comp…

Let $G$ be a cyclic group of order $p^r$ over a field ${\bf F}$ of characteristic $p$, let $V$ be the relevant finite-dimensional ${\bf F}G$-module, and write ${\bf F}[V]^G$ for it…

For an integer $n\geqslant2$, let $\rho(n)$ denote the smallest prime number that does not divide $n$. For the cyclic group $\mathbb{Z}_{\!/n}$, let $\lambda(\mathbb{Z}_{\!/n})$ de…

Let $G$ be a cyclic group, and let $\bar{A}\oplus\bar{B}=G$ be a direct-sum factorization. A factorization is quasiperiodic when its summands have the periodic structure specified…

Let ${\cal T}={\mathbb Z}+\frac{1}{m}A$, where $A\subseteq{\mathbb Z}$ is reduced modulo $m$ and admits a factorization $A\oplus B={\mathbb Z}/m{\mathbb Z}$. A universal spectrum f…

Let $\mathbb{Z}_n=\{0,1,\dots,n-1\}$ be the finite cyclic group with normalized counting measure, and let $(P_t)_{t\in\mathbb{R}_+}$ be the Poisson-like semigroup defined by … wher…

Let $T$ be a strong starter of odd order $p\geq 7$ coprime with $3$, let $\hat{T}$ be the set of pair sums of $T$, and let $t\in\mathbb{Z}_n^*\setminus\hat{T}$. Modular Sudoku solv…

Let $a_1,\dots,a_d$ be distinct nonzero elements of $\mathbb{Z}_p$, where $p$ is prime, and suppose that … A rearrangement is an ordering $a_{i_1},\dots,a_{i_d}$ whose partial sums…

Let $G$ be a finite group with a cyclic subgroup $C$ of index $n$, and let … be its normal core. The Babai–Goodman–Pyber core-index conjecture says that this core has index at most…

The high-density $3k-4$ conjecture. There are arithmetic progressions $P_A,P_B,P_C\subseteq G$, all having a common difference, such that

Let $\mathbb{Z}_N$ be the cyclic group of order $N$, and let $A\subset\mathbb{Z}_N$ be $\{x,2x,\ldots,(d-1)x\}$-free, meaning that no $x$ has all of $x,2x,\ldots,(d-1)x$ in $A$. Cy…

Cyclic cube-free density conjecture. If $d\mid N$, then

Let $n=2M+1$ be a positive odd number, and let $d_1,d_2,\dots,d_M$ be elements of $\mathbb{Z}_n\setminus\{0\}$. Pasotti–Pellegrini conjecture. The set $\mathbb{Z}_n\setminus\{0\}$…

Let $n$ be an integer with $n>11$, and let $k$ be an index parameter. The cyclic group of order $n$ is denoted by $4\mathbb{Z}_n$4$. **The cyclic-group$k…

Generalized Maximum Modulus Principle. There exists $C(d,N)>0$, independent of $n$, such that for every $n\geq 1$,