Using inequalities between rational numbers to solve systems of two exponential Diophantine equations (Q2883272)

In this note, the author presents a construction of a definite type of rational numbers \(p^m/q^n\) where \(p, q \in \mathbb{N}, p > 1, q > 1, \gcd(p, q) = 1\) between two given rational numbers \(a/b < c/d\) and for these numbers, he provides a unique pair of coprime positive integers \(u, v\) such that NEWLINE\[NEWLINE au^{1/k} + cv^{1/k} = (bc-ad)p^m, NEWLINE\]NEWLINE NEWLINE\[NEWLINE bu^{1/k} + dv^{1/k} = (bc-ad)q^n. NEWLINE\]NEWLINE The construction is an extension of the one of the author's in [JP J. Algebra Number Theory Appl. 23, No. 2, 165--170 (2011; Zbl 1237.11017)]. As an application, infinitely many integer solutions \(x, y, u, v\) are given for the system of Diophantine equations NEWLINE\[NEWLINE au^{1/k} + cv^{1/k} = (bc-ad)p^x, NEWLINE\]NEWLINE NEWLINE\[NEWLINE bu^{1/k} + dv^{1/k} = (bc-ad)q^y, NEWLINE\]NEWLINE where \(a, b, c, d, k \in \mathbb{N}\) and \(bc-ad > 0\).