14 problems

For an irrational real number $\alpha=[a_0;a_1,a_2,\ldots]$, let $B(\alpha)=\limsup_{n\to\infty}a_n(\alpha)$, where $a_n(\alpha)$ are its continued-fraction partial quotients. The…

For an irrational real number $\alpha=[a_0;a_1,a_2,\ldots]$, let $M(\alpha)=\sup_{n\geq 1}a_n(\alpha)$, where $a_n(\alpha)$ are its continued-fraction partial quotients. The 2-adic…

Conjecture 4. For all $k\geq 0$, the residue of $L_0(m)$ modulo $2^k$ is periodic with period dividing $2^k$; likewise, the residue of $L_1(m)$ modulo $2^k$ has period dividing…

Let $M(n)$, for $n\geq 0$, be the number of tilings of the Aztec diamond of order $n$ using dominos and square tetrominos. Conjecture 2. For all $k\geq 1$, if … then … This is pres…

Let $M(n)$, for $n\geq 0$, be the number of tilings of the Aztec diamond of order $n$ using dominos and square tetrominos. Conjecture 1. For all $k\geq 1$, the residue of $M(n)$ mo…

Let $5\le\Delta\le8$ and $e\ge5$. For $j\ge0$, define … For $j>0$, $j\ne2^{e-1}$, and any integer $m$, define … Auxiliary valuation conjecture. The quantities satisfy all the follo…

Let $5\le\Delta\le8$, $0\le m<2^{e-1}$, and $e\ge4$. If … then the difference-valuation conjecture. For every $d\ge0$, … This is presented as a reduction step toward the one-zero c…

Let $5\le\Delta\le8$, let $z_{\Delta,\ell}\in\mathbb Z_2$ be the two zeros of $P_\Delta$ indexed by $\ell\in\{0,1\}$, and let $\varepsilon_{\Delta,\ell}$ be $2$ for…

Let $e$ be a positive integer. The basic negative-argument valuation conjecture. … The paper presents this as the simplest case of its negative-argument congruence conjecture, givi…

Let $P_n(x)$ denote the partial Stirling function, let $\nu$ be the 2-adic valuation, and let $\operatorname{o}(n)=n/2^{\nu(n)}$ be the odd part of $n$. For integers $e$ and…

Let $e=\operatorname{lg}(n-1)$, and consider the congruence classes represented by $P_n(2^{e-1}x+p)$. The paper identifies several families in which a unique term indexed by…

Let $x\in\mathbb Z_2$ be a finite element, and let $i_0$ and $d$ be positive integers such that $x_{i_0}=0$ and $x_{i+d}=x_i$ for all $i\ge i_0$, with $2^{i+d}\le x$. Define … the…

Let $x\in\mathbb Z_2$ have binary digits $x_i\in\{0,1\}$, so that $x=\sum_{i\ge0}x_i2^i$. Suppose that, for some $d\ge2$ and $i_0\ge0$, $x_{i+d}=x_i$ for all $i\ge i_0$. The eventu…

Let $P_n$ be the partial Stirling function and let $u$ denote the exponent of $2$ in an integer. Suppose that, on a congruence class, $nu(Pn(x))=nu(x-x0)+c0$ for some…