3 problems

Let $\mathcal{P}$ be the positive integers equipped with the relation $\prec$ defined by $m\prec n$ if $\Phi_m(q)\leqslant\Phi_n(q)$ for every integer $q\geqslant 2$ and $m\ne n$.…

Let $\mathcal{P}$ be the set of positive integers, and let $\Phi_d(t)$ be the $d$th cyclotomic polynomial. Define a relation $\preceq$ on $\mathcal{P}$ by … and define $m\prec n$ w…

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…