Perfect powers as sums of convergent denominators of quadratic irrationals
Divyum Sharma, L. Singhal
TL;DR
The paper studies the Diophantine problem of bounding powers $y^a$ that arise as sums of convergent denominators $q_N$ of a fixed real quadratic irrational $\alpha$, i.e., $y^a = q_{N_1}+\cdots+q_{N_K}$. It develops two effective upper bounds for $y^a$ depending on the Hamming weight of $y$ in Zeckendorf (and more generally Ostrowski) representations and in radix representations, extending recent results of Vukusic–Ziegler and Kebli–Kihel–Larone–Luca. The authors prove finiteness and provide explicit, effectively computable bounds by splitting the convergent-denominator sequence into subsequences with the same recurrence and applying Baker-type estimates for linear forms in logarithms (Matveev), together with non-vanishing results for the associated linear forms. These results advance the understanding of perfect powers as sums of elements from linear recurrence sequences and yield two complementary, weight-based bounds with potential for explicit enumeration. The methods bridge Ostrowski numeration, Zeckendorf representations, and Baker theory to yield effective control over solutions in this Diophantine setting.
Abstract
Let $α$ 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 $α$. 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.