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

TL;DR

This paper investigates systems of Diophantine equations with many integer solutions, extending previous work by defining new systems that have solutions growing faster than known sequences as the number of variables increases.

Contribution

It introduces new systems of Diophantine equations with solutions that grow faster than previously known sequences, expanding understanding of solution counts in such systems.

Findings

Defined systems T_n with solutions t_n growing faster than b_n

Established that lim_{n o fty} t_n/b_n= finite

Extended previous results on solutions of Diophantine systems

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.