Let G(k) be defined as G(k) = & lang;a,b;a(-1)ba=b(k)& rang;, where k &NOTEQUexpressionL; 0. It is known that if p is a prime number then G(k) is residually a finite p-group iff p|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 iff (k, q) = 1, p|q - 1 and the order of k in the multiplicative group of the field 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 ?? k - 1, q ?? k - 1, (k, q) = 1, p|q - 1, the order of k modulo q is a p-number, and p and q are chosen among the first x primes. We state that, if 2 <= |k| <= 10 000 and 1 <= x <= 50 000, 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)|2 <= |k| <= 10 000} subset of (0.28; 0.31]. In addition, the dependence of the value f(k)(50 000) on k is investigated and an effective algorithm for checking a two-element set of prime numbers for compliance with the conditions of the last criterion is proposed. The results may have applications in the theory of computational complexity and algebraic cryptography.
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