Representing an integer and its powers in two unrelated number systems

TL;DR

This paper establishes upper bounds for powers of integers expressed as sums of convergent denominators of a quadratic irrational, linking these bounds to different number system representations and extending recent related results.

Contribution

It introduces new effective upper bounds for integer powers in terms of their representations in two unrelated number systems, extending prior work by Vukusic and Ziegler.

Findings

Derived bounds depend on Hamming weights in radix and Zeckendorf representations.

Extended a recent theorem by Vukusic and Ziegler.

Provided analogues of existing theorems in the context of quadratic irrationals.

Abstract

Let $\alpha$ 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 $(q_N)_{N\,\geq\,0}$ is the sequence of convergent denominators to $\alpha$. 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.