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Solutions to $xyz = 1$ and $x+y+z = k$ in algebraic integers of small degree, I

Volume 162 / 2014

H. G. Grundman, L. L. Hall-Seelig Acta Arithmetica 162 (2014), 381-392 MSC: Primary 11D25; Secondary 11G05, 11R16. DOI: 10.4064/aa162-4-5


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 $\mathbb {Q}$.


  • H. G. GrundmanDepartment of Mathematics
    Bryn Mawr College
    Bryn Mawr, PA 19010, U.S.A.
  • L. L. Hall-SeeligDepartment of Mathematics
    Merrimack College
    North Andover, MA 01845, U.S.A.

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