On systems of Diophantine equations with a large number of integer solutions

Abstract

Let E_n={x_i+x_j=x_k, x_i \cdot x_j=x_k: i,j,k \in {1,...,n}}. For each integer n \geq 13, J. Browkin defined a system B_n \subseteq E_n which has exactly b_n solutions in integers x_1,...,x_n, where b_n \in N\{0} and the sequence {b_n}_{n=13}^\infty rapidly tends to infinity. For each integer n \geq 12, we define a system T_n \subseteq E_n which has exactly t_n solutions in integers x_1,...,x_n, where t_n \in N\{0} and lim_{n \to \infty} t_n/b_n=\infty.