## Abstract

Let p be a rational prime and D a positive rational integer coprime with p. Denote by N (D,1, p) the number of solutions (x, n) of the equation Dx ^{2} + 1 = p^{n} in rational integers x ≥ 1 and n ≥ 1. In a paper of Le, he claimed that N (D, 1, p) ≤ 2 without giving a proof. Furthermore, the statement N(D, 1, p) ≤ 2 has been used by Le, Bugeaud and Shorey in their papers to derive results on certain Diophantine equations. In this paper we point out that the statement N (D,1, p) ≤ 2 is incorrect by proving that N (2, 1, 3) = 3.

Original language | English |
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Pages (from-to) | 3643-3645 |

Number of pages | 3 |

Journal | Proceedings of the American Mathematical Society |

Volume | 131 |

Issue number | 12 |

DOIs | |

State | Published - Dec 2003 |

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