Computational Analysis of Quantitative Characteristics of Some Residual Properties of Solvable Baumslag–Solitar Groups

Let Gk be defined as = \(\left\langle {a,b;{{a}^{{ - 1}}}ba {{b}^{k}}} \right\rangle \), where k ≠ 0. It is known that if p a prime number then residually finite p-group iff p|k – 1. also and q are primes not dividing 1, < π {p, q} π-group (k, q) p|q 1 the order of in multiplicative group field ℤq p-number. This paper examines question two-element sets numbers satisfy conditions last criterion. More precisely, let fk(x) such q, \(\nmid \) modulo p-number, chosen among first x primes. We state that, 2 ≤ |k| 10 000 50 000, for almost all considered function can approximated quite accurately by αkx0.85, coefficient αk different each {αk|2 000} ⊆ (0.28; 0.31]. In addition, dependence value fk(50 000) on investigated an effective algorithm checking set compliance with criterion proposed. The results may have applications theory computational complexity algebraic cryptography.