Diophantine solutions to systems of two linear functions made powers or exponentials (Q2900199)

In this note, the author presents infinitely many integer solutions first to the system of Diophantine equations NEWLINE\[NEWLINE a_1 u_1 + a_2 u_2 + \dots + a_N u_N = x^n NEWLINE\]NEWLINE NEWLINE\[NEWLINE b_1 u_1 + b_2 u_2 + \dots + b_N u_N = y^{mn} NEWLINE\]NEWLINE and second to NEWLINE\[NEWLINE a_1 u_1 + a_2 u_2 + \dots + a_N u_N = p^x NEWLINE\]NEWLINE NEWLINE\[NEWLINE b_1 u_1 + b_2 u_2 + \dots + b_N u_N = q^y NEWLINE\]NEWLINE where \(a_i \in \mathbb{Z}^*, b_i \in \mathbb{N}^* \;(i=1,2, \dots, N)\) such that \(a_1 / b_1 < a_2 / b_2 < \dots < a_N / b_N\) and \(a_i b_j - a_j b_i = -1 \;(a_i, a_j \in \mathbb{N}^*)\) for at least one pair \((i,j)\), further, \(m,n \in \mathbb{N}^*\), and \(p>1,q>1\) are coprime positive integers. In the proofs, the author applies real analysis and related own results. Furthermore, in two particular cases of systems of equations of the above shapes, it is demonstrated how the main results of this note can be used to find an integer solution to the corresponding system.