Publication: Integers of a quadratic field with prescribed sum and product
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Date
2023-03-01
Authors
Authors
Bremner, Andrew
Soydan, Gökhan
Journal Title
Journal ISSN
Volume Title
Publisher
Ars Polona-ruch
Abstract
For 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.
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Keywords
Xyz = x+y+z, Unit solutions, Equation, Science & technology, Physical sciences, Mathematics