On the Diophantine equation 2m + nx2 = yn

被引：7

作者：

Luca, Florian ^{[2
,3
]}

Soydan, Gokhan ^{[1
]}

机构：

[1] Isiklar Air Force High Sch, TR-16039 Bursa, Turkey

[2] Univ Nacl Autonoma Mexico, Inst Matemat, Morelia 58089, Michoacan, Mexico

[3] Univ Witwatersrand, John Knopfmacher Ctr Applicable Anal & Number The, ZA-2050 Po Wits, South Africa

来源：

JOURNAL OF NUMBER THEORY | 2012年 / 132卷 / 11期

关键词：

Exponential Diophantine equations; Primitive divisors of Lehmer sequences;

D O I：

10.1016/j.jnt.2012.05.010

中图分类号：

O1 [数学];

学科分类号：

0701 ; 070101 ;

摘要：

In this note, we prove that the Diophantine equation 2(m) + nx(2) = y(n) in positive integers x, y, m, n has the only solution (x, y,m,n) = (21,11,3,3) with n > 1 and gcd(nx, y) = 1. In fact, for n = 3,15, we transform the above equation into several elliptic curves for which we determine all their {2}-integer points. For n not equal 3,15, we apply the result of Yu.F. Bilu, G. Hanrot and P.M. Voutier about primitive divisors of Lehmer sequences. (C) 2012 Elsevier Inc. All rights reserved.

引用

页码：2604 / 2609

页数：6

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