Pillai's problem with Padovan numbers and powers of two

Ana María García Lomeli,Santos Hernández Hernández

doi:10.15446/recolma.v53n1.81034

Ana María García Lomeli, Santos Hernández Hernández

Open Access

https://doi.org/10.15446/recolma.v53n1.81034

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Abstract

Let (Pn)n≥0 be the Padovan sequence given by P0 = 0, P1 = P2 = 1 and the recurrence formula Pn+3 = Pn+1 + Pn for all n ≥ 0. In this note we study and completely solve the Diophantine equation Pn - 2m = Pn1 - 2m1 in non-negative integers (n, m, n1, m1).

Highlights

Let a, b be two fixed positive integers and consider the Diophantine equation an − bm = an1 − bm1 (1)ANA CECILIA GARCıA LOMELı & SANTOS HERNA NDEZ HERNA NDEZ in positive integers (n, m, n1, m1) with (n, m) = (n1, m1)
In [14], Pillai conjectures that in the case (a, b) = (2, 3) all solutions of equation (1) are (3, 2, 1, 1), (5, 3, 3, 1) and (8, 5, 4, 1). This conjecture remained open for about 37 years and it was confirmed by Stroeker and Tijdeman in [17] by using Baker’s theory on linear forms in logarithms of algebraic numbers
We study another particular case of equation (2) namely with Padovan numbers and powers of 2

Summary

Introduction

1. Introduction Let a, b be two fixed positive integers and consider the Diophantine equation an − bm = an1 − bm1 (1) Let U := (Un)n 0 and V := (Vm)m 0 be two linearly recurrence sequences of integers and look at the diophantine equation We study another particular case of equation (2) namely with Padovan numbers and powers of 2.

Results

Conclusion