Representing an integer and its powers in two unrelated number systems

Divyum Sharma, L. Singhal

Let $\hat{I}\pm$ be a fixed quadratic irrational. Consider the Diophantine equation \[ y^a\ =\ q_{N_1} + \cdots + q_{N_K},\quad N_1 \geq \cdots \geq N_{K} \geq 0,\quad a, y \geq 2 \] where…