On some exponential Diophantine equations. (English) Zbl 1014.11023

Bugeaud, Yann

Monatsh. Math. 132, No. 2, 93-97 (2001).

Let \(D_1,D_2\) be coprime positive integers, and let \(h\) denote the class number of the quadratic field \(\mathbb{Q} (\sqrt{-D_1D_2})\). In this paper, using a deep theorem concerning the existence of primitive divisors of Lucas and Lehmer numbers, the authors completely determine all solutions \((x,y,n)\) of the generalized Ramanujan-Nagell equations with the type \(D_1x^2+ D_2= \lambda^2y^n\), \(x,y,n\in \mathbb{N}\), \(y>1\), \(n>2\), \(\gcd(D_1x,D_2y)= \gcd(n,h)= 1\), \(\lambda\in \{1,2\}\).

Reviewer: Le Maohua (Zhanjiang)

MSC:

11D61

Exponential Diophantine equations

Keywords:

exponential Diophantine equation; generalized Ramanujan-Nagell equations