### Abstract

In this note we consider diophantine equations of the form. a(xp-yq)=b(zr-ws),where 1p+1q+1r+1s=1, with even positive integers p, q, r, s. We show that in each case the set of rational points on the underlying surface is dense in the Zariski topology. For the surface with (p, q, r, s) = (2, 6, 6, 6) we prove density of rational points in the Euclidean topology. Moreover, in this case we construct infinitely many parametric solutions in coprime polynomials. The same result is true for (p, q, r, s) ∈ {(2, 4, 8, 8), (2, 8, 4, 8)}. In the case (p, q, r, s) = (4, 4, 4, 4), we present some new parametric solutions of the equation x4-y4=4(z4-w4).

Original language | English (US) |
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Pages (from-to) | 46-64 |

Number of pages | 19 |

Journal | Journal of Number Theory |

Volume | 136 |

DOIs | |

State | Published - Mar 1 2014 |

### Keywords

- Diagonal diophantine equation
- Quartic surface
- Zariski topology

### ASJC Scopus subject areas

- Algebra and Number Theory

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## Cite this

Bremner, A., & Ulas, M. (2014). On certain diophantine equations of diagonal type.

*Journal of Number Theory*,*136*, 46-64. https://doi.org/10.1016/j.jnt.2013.09.008