Glasgow Mathematical Journal

Table of Contents - May 2012 - Volume 54, Issue 02  

Research Article

ON THE DIOPHANTINE EQUATION x2 + d2l + 1 = yn

ATTILA BÉRCZESa1 and ISTVÁN PINKa2

a1 Institute of Mathematics, University of Debrecen, Number Theory Research Group, Hungarian Academy of Sciences and University of Debrecen, H-4010 Debrecen, P.O. Box 12, Hungary e-mail: berczesa@math.klte.hu

a2 Institute of Mathematics, University of Debrecen, H-4010 Debrecen, P.O. Box 12, Hungary e-mail: pinki@math.klte.hu

Abstract

Let d > 0 be a squarefree integer and denote by h = h(−d) the class number of the imaginary quadratic field $\mathbb{Q}(\sqrt{-d})$. It is well known (see e.g. [25]) that for a given positive integer N there are only finitely many squarefree d's for which h(−d) = N. In [45], Saradha and Srinivasan and in [28] Le and Zhu considered the equation in the title and solved it completely under the assumption h(−d) = 1 apart from the case d ≡ 7 (mod 8) in which case y was supposed to be odd. We investigate the title equation in unknown integers (x, y, l, n) with x ≥ 1, y ≥ 1, n ≥ 3, l ≥ 0 and gcd(x, y) = 1. The purpose of this paper is to extend the above result of Saradha and Srinivasan to the case h(−d) ∈ {2, 3}.

(Received September 29 2010)

(Accepted September 24 2011)

2000 Mathematics Subject Classification

  • 11D41;
  • 11D61