Solutions to \(xyz=x+y+z=1\) in quintic number rings (Q2883395)

The equation in title was first considered by Cassels in 1960's who proved that it has no rational solutions. Due to several authors the solutions are known in the rings of integers of all quadratic, cubic and quartic number fields. In many respects degree five counts as higher degree for number fields. Despite of it the authors manage to list all quintic number fields for which the ring of integers contain solutions of this equation. These are three specific fields and four infinite parametric families of quintic number fields. For all these fields a triple of solutions is also given.