Prime ideal divisors of parametric recurrence sequences

TL;DR

This paper investigates prime ideal divisors of parametric linear recurrence sequences at roots of unity, providing bounds on prime divisors, the $S$-part, and finiteness results for related Diophantine equations.

Contribution

It introduces new bounds and finiteness results for prime ideal divisors and $S$-parts of specialized recurrence sequences at roots of unity.

Findings

Exponential lower bounds for prime ideal divisors.

Effective upper bounds for the $S$-part of the sequence.

Finiteness results for certain Diophantine equations.

Abstract

We prove new arithmetic results for parametric linear recurrence sequences specialized at roots of unity, denoted by $(U_n(\zeta))_{n\geq 0}$. In particular, we obtain exponential lower bounds for the largest prime ideal divisor and norm of the radical of the principal ideal generated by $U_n(\zeta)$. We further derive an effective upper bound for the $S$-part of $U_n(\zeta)$, showing that it is strictly smaller than a fixed power of its absolute norm for sufficiently large $n$. Finally, we establish an effective finiteness result for Diophantine equations of the form $U_n(\zeta)=w_1+\cdots+w_r,$ with $S$-unit variables.