## Abstract

Let A, B ∈ N with A > 1, B > 1 and gcd(A, B) = 1, k ≥ 2 be an integer coprime with AB, and let λ ∈ {1, 2, 4} be such that if λ = 4, then A ≠ 4 and B ≠ 4; and if k is even, then λ = 4. In this paper, we shall describe all solutions of the equation AX^{2} + BY^{2} = λk^{Z}, X, Y, Z ∈ Z, gcd(X, Y) = 1, Z > 0 with X|*A or Y |*B, where the symbol X|*A means that every prime divisor of X divides A. Then, using this result, we give some more general results on the number of solutions of the equation la^{x} + mb^{y} = λc^{z}, x > 1, y > 1, z >1. In addition, using Cao's resulton Pell equation, we obtain some improvement of Terai's results on the equations a^{x} + 2 = c^{z}, a^{x} + 4 = c^{z} and a^{x} + 2^{y} = c^{z}.

Original language | English |
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Pages (from-to) | 1015-1034 |

Number of pages | 20 |

Journal | Taiwanese Journal of Mathematics |

Volume | 12 |

Issue number | 5 |

DOIs | |

Publication status | Published - Aug 2008 |

## Scopus Subject Areas

- Mathematics(all)

## User-Defined Keywords

- Exponential diophantine equation
- Quadratic field