On a problem of Pillai involving S-units and Lucas numbers

TL;DR

The paper advances the Pillai-type study for Lucas numbers by solving $L_n-2^x3^y=c$ across regimes $c=0$, $c>0$, and $c<0$. It integrates the Binet representation, lower bounds for linear forms in logarithms, LLL lattice reduction, and $S$-unit techniques to derive effective bounds on $n$, $x$, and $y$, enabling complete classifications in the $c=0$ and $c>0$ cases and a finite, computable set for $c<0$. The authors enumerate explicit solutions for all qualifying $c$ values and provide rigorous bounds that reduce the problem to finite computational checks, combining analytic number theory with computer-assisted verification. This work extends Pillai-type results to Lucas sequences and $S=\\{2,3\ o$-units, illustrating how deep Diophantine tools yield concrete, enumerated solutions with potential applications to related exponential equations.

Abstract

Let $ \{L_n\}_{n\geq 0} $ be the sequence of Lucas numbers. In this paper, we look at the exponential Diophantine equation $L_n-2^x3^y=c$, for $n,x,y\in \mathbb{Z}_{\ge0}$. We treat the cases $c\in -\mathbb{N}$, $c=0$ and $c\in \mathbb{N}$ independently. In the cases that $c\in \mathbb{N}$ and $c\in -\mathbb{N}$, we find all integers $c$ such that the Diophantine equation has at least three solutions. These cases are treated independently since we employ quite different techniques in proving the two cases.

Theorems & Definitions (35)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Lemma 2.1: Lemma 1 in VZ
• Definition 2.1
• Theorem 2.1: Matveev, MAT
• Theorem 2.2: Laurent et al., LMN
• Definition 2.2
• Lemma 2.2: Bugeaud and Laurent, BL
• Lemma 2.3: Lemma 5 in VZ