22 problems

Let ${\boldsymbol{s}}\in\mathbb N^l$. Let $RJ({\boldsymbol{s}})$ be the reserved divisibility set, and let $RJ_1({\boldsymbol{s}})$ and $RJ_2({\boldsymbol{s}})$ denote the correspo…

Let $l$ be a positive integer and let ${\boldsymbol{s}}\in\mathbb N^l$. Define $J({\boldsymbol{s}}|2)$ to be the set of nonnegative integers $n$ for which the numerator of the mult…

Let $r\in\mathbb{N}_0$ and $n\in\mathbb{N}$. For any reals $x,y$, let ${\bf p}_{r+1}:=(p_1,p_2,\ldots,p_{r+1})\in\mathbb{N}^{r+1}$ and…

Pilehrood–Pilehrood–Tauraso's conjecture. For any positive integer $n$ with $n>k$, one has

Bernoulli congruences for $R_8^{(m)}(p)$, $R_9^{(m)}(p)$, and $R_{10}^{(m)}(p)$.

Cyclic-sum conjecture. For every integer $n>r$ and every primitive root of unity $\zeta_n$, the following hold. With $r=\sum_{j=0}^{t}d_j+2t$,

Let $k$ be a positive integer and let $p_1$ be a prime greater than $k+1$. For each index $\boldsymbol{k}$, let $H_{p-1}(\boldsymbol{k})$ be the multiple harmonic sum, and let…

For $n_{1},\dots,n_{d}\in\mathbb{Q}$, define … and let $W_{k}^{\star}(n_{1},\dots,n_{d})$ be the analogous sum with $\zeta_{\mathcal{A}}^{\star}$. For $k,r\in\mathbb{Z}_{\geq1}$ an…

Let $W_{k}(n_{1},\dots,n_{d})$ and $W_{k}^{\star}(n_{1},\dots,n_{d})$ denote the weighted sums of finite multiple zeta values and finite star multiple zeta values, respectively, de…

Define the weighted sums of finite multiple zeta values by … and define $W_{k}^{\star}(n_{1},\dots,n_{d})$ similarly with $\zeta_{\mathcal{A}}^{\star}$ in place of…

Let $m,n\in\mathbb{N}$, and let $\gb_a$ denote the $\mathcal{A}$-Bernoulli numbers. Write $a_1+\dots+a_l\vdash n$ for the summation condition used in the source, and let…

Let $\mathbb{N}$ denote the positive integers. For each $m,n\in\mathbb{N}$, let $R_n^{(m,1)}$ and $S_n^{(m,1)}$ be the finite multiple harmonic sum quantities defined in the source…

Consider the domains and summands described in the source: domains are formed from finite disjoint unions and finite products of sets defined by chains of equalities and inequaliti…

Consider convergent sums of multiple harmonic values that are both $p$-adically and weight-adically convergent. Generation conjecture for multiple harmonic-value relations. Every r…

Let $I\subset\mathcal{P}\times\mathbb{N}$, and let $\widehat{\operatorname{per}}_{\operatorname{har}_I}$ be the period map on weight-adically convergent sums of motivic multiple ha…

Weighted congruence deduction conjecture. Every weighted congruence can be deduced from Theorem.

Fix $N>1$ and work on $\mathbb{P}^{1}-\{0,\mu_N,\infty\}$. Let $p$ range over primes not dividing $N$, let $\alpha\in\mathbb{N}$, and let $\operatorname{har}_{p^{\alpha}}$ denote t…

Let $\mathcal{P}_{\geq 3}$ denote the set of primes at least $3$, let $\alpha\in\mathbb{N}^{\ast}$, and let $I$ be one of $\mathcal{P}_{\geq 3}\times\{\alpha\}$,…

Reserved density conjecture.

Reserved-set structure conjecture. If one of the following holds: (i) ${\vec{s}}=1$, (ii) ${\vec{s}}=(1,2,1)$, (iii) ${\vec{s}}=(2r-1,1)$ for some $r\geq1$, or (iv)…

Conjecture on 2-divisible sets. For every positive integer $d$ and every ${\vec{s}}\in{\mathbb N}^d$,

Finiteness conjecture. The set $J({\vec{s}}|p)$ is finite for every prime $p$.