26 problems

A Markov number is a positive integer appearing in a solution to the Markov equation … A triple satisfying this equation is a Markov triple. Frobenius' uniqueness conjecture. Every…

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 $(p,q)\in\mathcal{R}$…

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

Gyoda–Maruyama's uniqueness conjecture. The Uniqueness Conjecture remains true for $k$-Markov numbers: every $k$-Markov number is the largest number in a unique corresponding $k$-M…

Let $p,q$ be coprime positive integers, and let $m_{q/p}$ denote the Markov number associated with $q/p$. Suppose that $0<i<q$ and $\gcd(p+i,q-i)=1$. Aigner's constant sum conjectu…

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…

Fix a Markov number $cell$ with $cell\geq 5$, and consider either of the two classical Markov-tree branches of triples $(cell,b,c)$ with $cell$ fixed. Let $m_{cell}(q)$ be the mirr…

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…

Uniqueness conjecture. Each Markov number has only one pair of companion numbers.

Let $t$ be a rational number, let $\tilde w_t(A,B)$ be the corresponding Christoffel word, and let $\hat A$ and $\hat B$ be the displayed $y$-deformed generators. Write … Let $F_t$…

For an integer $\ell$, let $\mathrm{P}\mathbb T(\ell)$ be the tree whose vertices are triples of integer column vectors as constructed in the source, and let…

A Markov triple is a positive integer solution $(a,b,c)$ of … A positive integer occurring in a Markov triple is called a Markov number. Aigner's conjecture. For any Markov number…

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…

A Markov number is a number appearing in a Markov triple generated by mutation. For an initial triple with distinct entries, the mutation tree can have distinct shadows associated…

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 $p$, $q$ and $i$ be positive integers such that $p<q$, $\gcd(q,p)=1$, and $\gcd(q+i,p)=1$. For each reduced positive rational number $r$, let $m_r$ denote the associated Markov…

Let $(d,n)\in(\mathbb{N}\setminus\{0\})^2$, and let $c(n,d)$ be the distinguished path associated with $(n,d)$; let $m$ assign an $m$-value to paths. A path avoids the factor bb' i…

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…

For each pair $(p,q)\in\mathbb{N}^2$, generalized Markov numbers are defined by the construction in the source, extending the usual case in which $p$ and $q$ are relatively prime.…

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…