Solutions to $xyz = 1$ and $x+y+z = k$ in algebraic integers of small degree, I
Volume 162 / 2014
Acta Arithmetica 162 (2014), 381-392
MSC: Primary 11D25; Secondary 11G05, 11R16.
DOI: 10.4064/aa162-4-5
Abstract
Let $k\in \mathbb {Z}$ be such that $|\mathcal E_k(\mathbb {Q})| = 3$, where $\mathcal E_k: y^2 = 1 - 2 k x + k^2 x^2 -4 x^3$. We determine all solutions to $xyz = 1$ and $x + y + z = k$ in integers of number fields of degree at most four over $\mathbb {Q}$.
