46 problems

Let $p$ be a prime, let $a$ be a positive integer with $(a,p)=1$, and let $k\geq3$ be odd. Write $s(N,K)$ for the Stirling number of the first kind and let $v_p$ denote the $p$-adi…

Let $\mathcal{alpha}$ be a real number, let $p$ be a prime, and write $|n|_p=p^{-\nu_p(n)}$ for the $p$-adic norm of a positive integer $n$. Let $|\langle x\rangle|$ denote the dis…

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…

Let $p$ be a prime, let $k$ be a positive integer, and let … be the $k$-th elementary symmetric function of $1,1/2,\ldots,1/n$. Let $v_p$ denote the $p$-adic valuation. Leonetti–Sa…

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…

Let $x+iy$ be any non-trivial zero of the Riemann zeta function, and let $p$ be any prime. The quantity $p^{iy}$ is the phase factor associated with the imaginary part of the zero.…

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 a $p$-adic algebraic number be represented as a $p$-adic Hahn series, and call its support bounded when it is bounded in the relevant ordered exponent group. Bounded-support fi…

Let $p$) be a prime and let $\mathbb{Z}_p$ denote the ring of $p$-adic integers. A $p$-adic integer is normal in base $p$ when its base-$p$ digit expansion has every finite block w…

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…

For a prime $p$, let $(T_n)$ be the Tribonacci sequence defined by … with $T_0=0$ and $T_1=T_2=1$, and let $up$ denote the $p$-adic valuation. Marques–Lengyel conjecture. The…

Gaussian localization conjecture. If $\pi$ is not real, this ring is an extension of $\mathbb Q_{N(\pi)}$; if $\pi\equiv3\bmod4$ is a real prime, it is the ring of $2\times2$ matri…

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

Let $p$ be a prime, and consider the continued fraction expansions produced by the new algorithms for quadratic irrationals in $\mathbb{Q}_p$. Periodicity conjecture. The continued…

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 $x$,

Composite-dilation invariance conjecture. The distribution of the zeta eigenvalues is unchanged under these dilations when $n$ is composite, for example when $n=6$.

Coprimality conjecture. The integer $q$ must be relatively prime to $p$; otherwise the distribution is uniform.

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 $S(n,k)$ denote a Stirling number of the second kind and let $bd_2$ be the $2$-adic valuation. Amdeberhan's conjecture. For every integer $h\ge 0$ and every integer $k$ with…

Let $S(n,k)$ denote a Stirling number of the second kind, let $u2$ be the $2$-adic valuation, let $c32(c)$ be the sum of the binary digits of $c$, and let $binom{n}{k}$ denote a b…