11 problems

Let $a_{11},a_{22},a_{33},a_{12},a_{13},a_{23},d$ be integers, and consider the generalized Markoff equation … Let $D$ and $A_{11},A_{22},A_{33}$ denote the quantities defined earl…

Baragar's conjecture. The Markoff graph over $\mathbb{F}_p$ is connected. This conjecture concerns the dynamics of Markoff triples over finite fields and was attributed to Baragar…

Fix $\kappa\in R$ and let $\mathscr{M}(R)\subseteq R^3$ be the set of Markoff triples satisfying the corresponding Markoff equation. Let $\Gamma$ be the group generated by the Viet…

Let $q$ be a prime power and let an essential triple $(x,y,z)\in\mathbb{F}_q^3$ be the image under $(A,B)\mapsto(\operatorname{tr}A,\operatorname{tr}B,\operatorname{tr}(AB))$ of a…

Positivity conjecture. Every triple $(A,B,C)$ of positive integer solutions to the shadow Markoff equation can be obtained from an initial triple with $(\alpha:\beta:\gamma)$ a rat…

Let $\mathcal{O}(m)$ denote the set associated in the paper with the orders of minimal triples for the natural number $m$. Divisibility-by-three conjecture. There are infinitely ma…

Let $m=p$ be a prime congruent to $1$ modulo $4$, and let minimal triples be the minimal solution triples associated with the generalized Markoff equation in the paper. Prime one-m…

Let $m$ range over the natural numbers, and let a minimal triple mean a minimal solution triple of the generalized Markoff equation considered in the paper. One-minimal-triple conj…

Let $m$ be a natural number, and consider the minimal triples associated with the generalized Markoff equation studied in the paper. Minimal-triple growth conjecture. For a given…

Let $\mathcal M$ be the set of natural-number solutions to … Let $M$ be the set of coordinates occurring in triples of $\mathcal M$, and let $M^s$ be the multiset of largest coordi…

Classification conjecture.