20 problems

Let $k$ be a global function field, let $w$ be any place of $k$, and let $N_w$ denote the non-classical set for the interpolation of $L(\chi,s)$ at $w$; when $w=\infty$, this inter…

Bounded-support conjecture. The support $\operatorname{Supp}(f)$ must be a finite set.

Let the support of a $p$-adic Hahn expansion be the ordered set of exponents with nonzero coefficients, and let its order type be the order type of that set. Kedlaya's prediction.…

Let $\alpha\in\overline{\mathbf{Q}}\setminus\mathbf{Q}$, and let $S$ be the set of rational primes that are totally split in $\mathbf{Q}(\alpha)$. Let $r$ and $s$ be integers, and…

Let $k$ be an even positive integer and let $e\geq 1$. Write $g_{2,e}(k)$ for the least number of $k$-th powers needed to represent every element of the ramified $2$-adic ring of r…

Let $(x_n)$ be a third-order linear recurrence sequence, let $p$ be a prime number, and let $up$ denote the $p$-adic valuation. Bilu et al.'s conjecture. There exists a positi…

Restricted-product conjecture. The ring of quasi-endomorphisms satisfies

The efficient-prime finite-control conjecture. For any $\alpha>\frac{2}{\log 2}$, there is a finite set of primes $S_\alpha$ such that, for any $m$, if $v_p$ is sufficiently large…

The balanced and efficient numbers counting conjecture. For all $x$,

Let $f(b,c)=\sigma_2(b)-\sigma_2(c)+\nu_2\left(\binom{2c-b-1}{c}\right)$, where $c3_2$ is the binary digit sum and $bd_2$ is the $2$-adic valuation. Let $a,b,c,h$ be positive integ…

Let $c$ be a positive integer, let $S(n,k)$ denote a Stirling number of the second kind, let $bd_2$ be the $2$-adic valuation, and let $c3_2(m)$ denote the sum of the binary digits…

Let $p$ be a prime and let $t\geq1$, excluding the case $p=3$, $t=1$. A Browkin continued fraction (BCF) is a $p$-adic continued fraction expansion, and $\mathcal{Y}$ is the set of…

Let $p$ be an odd prime, let $\mathcal{Y}$ be the set of allowed partial quotients, and let $a_0,\ldots,a_{t-2}\in\mathcal{Y}$ satisfy $a_i>0$ and $|a_0|_\infty<\frac p4$. Extensio…

Let $p$ be an odd prime, let $\mathcal{Y}$ denote the set used in the paper for the allowed partial quotients, and let a nice BCF be a finite Browkin continued fraction satisfying…

Let $\zeta\in\mathbb{Z}_2$ be the $2$-adic number defined in Equation, and let $\ell_\zeta(a)$ denote the length of the zero block beginning at the relevant digit position indexed…

Let $\mathbb{Z}_{p}$ be the ring of $p$-adic integers. Fix distinct $d_{1},\dots,d_{r}\in\mathbb{Z}_{\geq 1}$ and not necessarily distinct $k_{1},\dots,k_{r}\in\mathbb{Z}_{\geq 0}$…

Let $p$ be an odd prime. Let $a,n,k$ be positive integers with $1\le a\le p-1$ and $2\le k\le a(p-1)p^{n-1}+1$. Define $\epsilon_k=0$ if $k$ is even and $\epsilon_k=1$ if $k$ is od…

Berrizbeitia–Medina–Moll–Moll–Noble conjecture. Fix a positive integer $k$ and a prime number $p$ such that $k<p$. Then there exist $m_0(k)\in\mathbb{N}_+$ and…

Amdeberhan–Manna–Moll conjecture. Fix a number $k\in\mathbb{N}_+$. Then there exist $m_0(k)\in\mathbb{N}_+$ and $\mu(k)\in\mathbb{N}$ such that for any integer $m\geq m_0(k)$ there…

2- and 3-adic expansion conjecture. The polynomials $\gamma_{-\frac{1}{2}+r}^{\ast\ast}(a)$ possess 2- and 3-adic expansions to all orders: