### Abstract

In this paper, the existence of effectively computable bounds on the solutions to the diophantine equation ap^{x}+bq^{y} = c+dp^{z}q^{w} is shown. In this equation p, q are taken to be fixed relatively prime positive integers and a, b, c, d positive integers. The methods involve the application of linear forms in both real and p-adic logarithms. Also, a result on an inequality involving S-integers is used. All constants involved can be explicitly computed in terms of the parameters a, b, c, d, p, q, conceivably allowing one to list all solutions to (^{*}) for any set of parameters. It is also indicated how the bounds in a particular case can be reduced to allow the practical solution of the equation. Finally, the methods are demonstrated by the solving of the equations 2^{x} + 3^{y} = 1 + 2^{z}3^{w} and 5 · 2^{x} + 7 · 3^{y} = 11 + 2^{z}3^{w}.

Original language | English (US) |
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Pages (from-to) | 194-207 |

Number of pages | 14 |

Journal | Journal of Number Theory |

Volume | 35 |

Issue number | 2 |

DOIs | |

State | Published - Jun 1990 |

### All Science Journal Classification (ASJC) codes

- Algebra and Number Theory