Publication: Integers of a quadratic field with prescribed sum and product
dc.contributor.author | Bremner, Andrew | |
dc.contributor.author | Soydan, Gökhan | |
dc.contributor.buuauthor | SOYDAN, GÖKHAN | |
dc.contributor.department | Bursa Uludağ Üniversitesi/Fen-Edebiyat Fakültesi/Matematik Bölümü | |
dc.contributor.researcherid | M-9459-2017 | |
dc.date.accessioned | 2024-09-30T10:46:50Z | |
dc.date.available | 2024-09-30T10:46:50Z | |
dc.date.issued | 2023-03-01 | |
dc.description.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. | |
dc.identifier.doi | 10.4064/cm9023-11-2022 | |
dc.identifier.eissn | 1730-6302 | |
dc.identifier.issn | 0010-1354 | |
dc.identifier.issue | 1 | |
dc.identifier.uri | https://doi.org/10.4064/cm9023-11-2022 | |
dc.identifier.uri | https://hdl.handle.net/11452/45497 | |
dc.identifier.volume | 173 | |
dc.identifier.wos | 000942494600001 | |
dc.identifier.wos | https://www.impan.pl/pl/wydawnictwa/czasopisma-i-serie-wydawnicze/colloquium-mathematicum/all/173/1/115046/integers-of-a-quadratic-field-with-prescribed-sum-and-product | |
dc.indexed.wos | WOS.SCI | |
dc.language.iso | en | |
dc.publisher | Ars Polona-ruch | |
dc.relation.journal | Colloquium Mathematicum | |
dc.relation.publicationcategory | Makale - Uluslararası Hakemli Dergi | |
dc.rights | info:eu-repo/semantics/closedAccess | |
dc.subject | Xyz = x+y+z | |
dc.subject | Unit solutions | |
dc.subject | Equation | |
dc.subject | Science & technology | |
dc.subject | Physical sciences | |
dc.subject | Mathematics | |
dc.title | Integers of a quadratic field with prescribed sum and product | |
dc.type | Article | |
dspace.entity.type | Publication | |
relation.isAuthorOfPublication | 356f7af9-3f0f-4c82-8733-d98627634647 | |
relation.isAuthorOfPublication.latestForDiscovery | 356f7af9-3f0f-4c82-8733-d98627634647 |