On the Diophantine equation $U_n-b^m = c$

TL;DR

This paper proves that for large enough base b, the Diophantine equation involving a fixed linear recurrence sequence and exponential terms has at most two solutions beyond a certain index, with applications to Tribonacci numbers.

Contribution

It establishes effective bounds and a finiteness result for solutions to the equation involving linear recurrence sequences and exponential terms, with explicit bounds for Tribonacci numbers.

Findings

At most two solutions for large b beyond a certain index

Explicit bounds for B and N_0 in the Tribonacci case

Implementation of the reduction algorithm in Sage

Abstract

Let $(U_n)_{n\in \mathbb{N}}$ be a fixed linear recurrence sequence defined over the integers (with some technical restrictions). We prove that there exist effectively computable constants $B$ and $N_0$ such that for any $b,c\in \mathbb{Z}$ with $b> B$ the equation $U_n - b^m = c$ has at most two distinct solutions $(n,m)\in \mathbb{N}^2$ with $n\geq N_0$ and $m\geq 1$. Moreover, we apply our result to the special case of Tribonacci numbers given by $T_1= T_2=1$, $T_3=2$ and $T_{n}=T_{n-1}+T_{n-2}+T_{n-3}$ for $n\geq 4$. By means of the LLL-algorithm and continued fraction reduction we are able to prove $N_0=1.1\cdot 10^{37}$ and $B=e^{438}$. The corresponding reduction algorithm is implemented in Sage.