PDF files of articles are only available for institutions which have paid for the online version upon signing an Institutional User License.

On a simple quartic family of Thue equations over imaginary quadratic number fields

Volume 208 / 2023

Benjamin Earp-Lynch, Bernadette Faye, Eva G. Goedhart, Ingrid Vukusic, Daniel P. Wisniewski Acta Arithmetica 208 (2023), 355-389 MSC: Primary 11D59; Secondary 11R11, 11Y50. DOI: 10.4064/aa230329-19-6 Published online: 8 September 2023


Let $t$ be any imaginary quadratic integer with $|t|\geq 100$. We prove that the inequality \[ |F_t(X,Y)| = | X^4 - t X^3 Y - 6 X^2 Y^2 + t X Y^3 + Y^4 | \leq 1 \] has only trivial solutions $(x,y)$ in integers of the same imaginary quadratic number field as $t$. Moreover, we prove results on the inequalities $|F_t(X,Y)| \leq C|t|$ and $|F_t(X,Y)| \leq |t|^{2 -\epsilon }$. These results follow from an approximation result that is based on the hypergeometric method. The proofs in this paper require a fair amount of computations, for which the code (in Sage) is provided.


  • Benjamin Earp-LynchCarleton University
    Ottawa, ON, Canada
  • Bernadette FayeUFR SATIC
    Université Alioune Diop de Bambey
    Bambey 30, Diourbel, Sénégal
  • Eva G. GoedhartFranklin & Marshall College
    Lancaster, PA 17603, USA
  • Ingrid VukusicUniversity of Salzburg
    5020 Salzburg, Austria
  • Daniel P. WisniewskiDepartment of Mathematics/Computer Science
    DeSales University
    Center Valley, PA 18034, USA

Search for IMPAN publications

Query phrase too short. Type at least 4 characters.

Rewrite code from the image

Reload image

Reload image