Bremner, AndrewSoydan, Gökhan2024-09-302024-09-302023-03-010010-1354https://doi.org/10.4064/cm9023-11-2022https://hdl.handle.net/11452/45497https://www.impan.pl/pl/wydawnictwa/czasopisma-i-serie-wydawnicze/colloquium-mathematicum/all/173/1/115046/integers-of-a-quadratic-field-with-prescribed-sum-and-productFor given k, $ is an element of Z we study the Diophantine systemx + y + z = k, xyz =lfor x, y, z integers in a quadratic number field, which has a history in the literature. When $ = 1, we describe all such solutions; only for k = 5, 6, do there exist solutions in which none of x, y, z are rational. The principal theorem of the paper is that there are only finitely many quadratic number fields K where the system has solutions x, y, z in the ring of integers of K. To illustrate the theorem, we solve the above Diophantine system for (k, $) = (-5, 7). Finally, in the case $ = k, the system is solved completely in imaginary quadratic fields, and we give (conjecturally) all solutions when $ = k <= 100 for real quadratic fields.eninfo:eu-repo/semantics/closedAccessXyz = x+y+zUnit solutionsEquationScience & technologyPhysical sciencesMathematicsArticle000942494600001173110.4064/cm9023-11-20221730-6302