14 problems

Let $\mathcal{R}$ be the set of labels used for generalized $k$-Markov numbers, and let $m^{(k)}_{(p,q)}$ denote the generalized $k$-Markov number attached to $(p,q)\in\mathcal{R}$…

Consider triples in the mirror deformation of the Markov tree, with each triple consisting of mirror deformed Markov numbers and with degree taken in the deformation parameter $q$.…

Let $m_n$ denote the mirror deformed Markov numbers. Positivity conjecture. Every coefficient of every $m_n$ is positive. The claim is based on experimental computations and is sta…

Fix a Markov number $\ell\geq 5$. Let $m_\ell(q)$ be a mirror deformation of $\ell$, and let $(p_n)_{n\geq 0}$ be the sequence of polynomials on a mirror Markov-tree branch fixing…

Aigner's conjectures. The following ordering properties should hold:

Let $(z_1,z_2,z_3)$ be an initial seed triple of roots of unity, and let $n$ be the minimal positive integer such that $z_i^n=1$ for each $i$. A triple of rational numbers…

Let $(z_1,z_2,z_3)$ be a triple of complex roots of unity, and let $n$ be the minimal positive integer such that $z_i^n=1$ for each $i$. Write the arguments as fractions of a full…

Let $k\in\mathbb Z_{\geq 0}$. A $k$-generalized Markov number is a number appearing in a $k$-generalized Markov triple. Uniqueness conjecture for generalized Markov triples. For an…

Let $q$ be a Markov number, and call a rational number a Markov fraction when it is one of the Markov fractions defined in the paper; its denominator is the denominator in lowest t…

For each pair $(q,p)in \mathbb Z_{>0}^2$ with $q\geq p$, let $m_{q,p}$ denote the generalized Markov number associated with $(q,p)$. Uniqueness conjecture for generalized Markov nu…

Let $c(n,d)$ be the path associated with $(n,d)$ in the generalized Markov construction, and let $m(c(n,d))$ denote its associated value. Unicity conjecture. If … then … Thus, with…

Let $a$ and $b$ be positive integers with $a<b$. The graph of general Markov numbers is the graph $\mathcal{T}\big((a,a),(b,b)\big)$, whose vertices are triples of integers constru…

Let $T_{\mu,\nu}$ be a generalised Markov triple-graph, and let a generalised Markov number be a number associated with this graph. A maximal (middle) element is the middle entry o…

Let $m_{p/q}$ denote the Markov number indexed by a reduced rational number $p/q$ with $0<p<q$. Fixed denominator conjecture. For positive integers $p,q,i$ such that $p+i<q$,…