Powers in products of terms of Pell's and Pell-Lucas sequences

被引：17

作者：

Bravo, Jhon J. ^{[1
]}

Das, Pranabesh ^{[2
]}

Guzman, Sergio ^{[3
]}

Laishram, Shanta ^{[2
]}

机构：

[1] Univ Cauca, Dept Matemat, Popayan, Colombia

[2] India Stat Inst, Stat Math Unit, New Delhi 110016, India

[3] Univ Nacl Autonoma Mexico, Ctr Ciencias Matemat, Morelia 58089, Michoacan, Mexico

来源：

INTERNATIONAL JOURNAL OF NUMBER THEORY | 2015年 / 11卷 / 04期

关键词：

Recurrence sequences; Pell sequences; Pell-Lucas sequences; arithmetic progressions; GREATEST PRIME DIVISOR; DIOPHANTINE EQUATIONS;

D O I：

10.1142/S1793042115500682

中图分类号：

O1 [数学];

学科分类号：

0701 ; 070101 ;

摘要：

In this paper, we consider the usual Pell and Pell-Lucas sequences. The Pell sequence (u(n))(n=0)(infinity) is given by the recurrence u(n) = 2u(n-1)+ u(n-2) with initial condition u(0) = 0, u(1) = 1 and its associated Pell-Lucas sequence (v(n))(n=0)(infinity) is given by the recurrence v(n) = 2v(n-1) + v(n-2) with initial condition v(0) = 2, v(1) = 2. Let n, d, k, y, m be positive integers with m >= 2, y >= 2 and gcd( n, d) = 1. We prove that the only solutions of the Diophantine equation u(n)u(n+d) ... u(n+(k-1)d) = y(m) are given by u(7) = 13(2) and u(1)u(7) = 13(2) and the equation v(n)v(n+d) ... v(n+(k-1)d) = y(m) has no solution. In fact, we prove a more general result.

引用

页码：1259 / 1274

页数：16

共 23 条

[1]

[Anonymous], 1990, GREATEST PRIME FACTO

[2] Ternary diophantine equations via galois representations and modular forms [J].