Let $G_{k}$ be defined as $G_{k} = \langle a, b;\ a^{-1}ba = b^{k} \rangle$, where $k \ne 0$. It is known that, if $p$ is some prime number, then $G_{k}$ is residually a finite $p$-group if and only if $p \mid k - 1$. It is also known that, if $p$ and $q$ are primes not dividing $k - 1$, $p < q$, and $\pi = \{p,\,q\}$, then $G_{k}$ is residually a finite $\pi$-group if and only if $(k, q) = 1$, $p \mid q - 1$, and the order of $k$ in the multiplicative group of the field $\mathbb{Z}_{q}$ is a $p$\-number. This paper examines the question of the number of two-element sets of prime numbers that satisfy the conditions of the last criterion. More precisely, let $f_{k}(x)$ be the number of sets $\{p,\,q\}$ such that $p < q$, $p \nmid k - 1$, $q \nmid k - 1$, $(k, q) = 1$, $p \mid q - 1$, the order of $k$ modulo $q$ is a $p$\-number, and $p$, $q$ are chosen among the first $x$ primes. We state that, if $2 \leq |k| \leq 10000$ and $1 \leq x \leq 50000$, then, for almost all considered $k$, the function $f_{k}(x)$ can be approximated quite accurately by the function $\alpha_{k}x^{0.85}$, where the coefficient $\alpha_{k}$ is different for each $k$ and $\{\alpha_{k} \mid 2 \leq |k| \leq 10000\} \subseteq (0.28;\,0.31]$. We also investigate the dependence of the value $f_{k}(50000)$ on $k$ and propose an effective algorithm for checking a two-element set of prime numbers for compliance with the conditions of the last criterion. The results obtained may have applications in the theory of computational complexity and algebraic cryptography.
The Inequalities of Merris and Foregger for Permanents