What question did this study set out to answer?

This research examines the solutions of Diophantine equations related to polynomial recursive sequences.

June 29, 2026

Diophantine equations for polynomial recursive sequences

Key Points

This research examines the solutions of Diophantine equations related to polynomial recursive sequences.
Analyzed Diophantine equations of a specific type involving polynomial power sums over a number field.
Applied the finiteness criterion of Bilu and Tichy to establish boundedness in solutions.
Demonstrated that certain Diophantine equations have infinitely many solutions with bounded denominators.
Showed that if the conditions for third-order linear recurrence sequences hold, then the sequences remain bounded.

Abstract

We study the Diophantine equation of type Formula: see text, where Formula: see text and Formula: see text are polynomial power sums defined over a number field Formula: see text. By applying the finiteness criterion of Bilu and Tichy, we show under appropriate assumptions that equation Formula: see text has infinitely many solutions with bounded Formula: see text-denominator. We also study decomposable polynomials in third and second order linear recurrence sequences. In particular, we show that if Formula: see text for a simple third order linear recurrence sequence Formula: see text of complex polynomials, then Formula: see text is bounded. Furthermore, we show that if Formula: see text for a binary recurrence sequence Formula: see text then Formula: see text is bounded.

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Cite This Study

Darsana et al. (Fri,) studied this question.

synapsesocial.com/papers/6a420b08f91bb43ea91923c9 https://doi.org/https://doi.org/10.1142/s179304212650123x

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