Representing an integer and its powers in two unrelated number systems

Abstract

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 $(q_N{)}_{N\ge 0}$ is the sequence of convergent denominators to $\hat{I}\pm$. We find two effective upper bounds for $y^a$ which depend on the Hamming weights of $y$ with respect to its radix and Zeckendorf representations, respectively. The latter bound extends a recent result of Vukusic and Ziegler. En route, we obtain an analogue of a theorem by Kebli, Kihel, Larone and Luca.

Modified version to appear in Acta Arithmetica