Non-trivial solutions of Aap + Bbp = Cc3 over number fields.
Saved in:
| Title: | Non-trivial solutions of Aap + Bbp = Cc3 over number fields. |
|---|---|
| Authors: | Kara, Yasemin1 (AUTHOR) yasemin.kara@bogazici.edu.tr, Nomden, Stef2 (AUTHOR) s.nomden@math.leidenuniv.nl, Özman, Ekin2 (AUTHOR) e.ozman@rug.nl |
| Source: | Journal of Number Theory. Sep2026, Vol. 286, p108-130. 23p. |
| Subjects: | Diophantine equations, Quadratic fields, Modular forms, Algebraic number theory, Mathematical bounds, Number theory |
| Abstract: | In this paper, we investigate solutions to the Diophantine equation A a p + B b p = C c 3 over number fields using the modular method. Assuming certain standard modularity conjectures, we first establish an asymptotic result for general number fields satisfying an appropriate S -unit condition. In particular, we verify that this condition holds for several imaginary quadratic fields. Beyond the asymptotic setting, we also obtain an effective result. Specifically, for the equation a p + d b p = c 3 over K = Q (− d) with d ∈ { 7 , 19 , 43 , 67 } , we determine an explicit bound (depending on d) such that no non-trivial solutions of a certain type exist whenever p exceeds this bound. [ABSTRACT FROM AUTHOR] |
| Copyright of Journal of Number Theory is the property of Academic Press Inc. and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.) | |
| Database: | Engineering Source |
| FullText | Text: Availability: 0 |
|---|---|
| Header | DbId: egs DbLabel: Engineering Source An: 193094113 AccessLevel: 6 PubType: Academic Journal PubTypeId: academicJournal PreciseRelevancyScore: 0 |
| IllustrationInfo | |
| Items | – Name: Title Label: Title Group: Ti Data: Non-trivial solutions of Aap + Bbp = Cc3 over number fields. – Name: Author Label: Authors Group: Au Data: <searchLink fieldCode="AR" term="%22Kara%2C+Yasemin%22">Kara, Yasemin</searchLink><relatesTo>1</relatesTo> (AUTHOR)<i> yasemin.kara@bogazici.edu.tr</i><br /><searchLink fieldCode="AR" term="%22Nomden%2C+Stef%22">Nomden, Stef</searchLink><relatesTo>2</relatesTo> (AUTHOR)<i> s.nomden@math.leidenuniv.nl</i><br /><searchLink fieldCode="AR" term="%22Özman%2C+Ekin%22">Özman, Ekin</searchLink><relatesTo>2</relatesTo> (AUTHOR)<i> e.ozman@rug.nl</i> – Name: TitleSource Label: Source Group: Src Data: <searchLink fieldCode="JN" term="%22Journal+of+Number+Theory%22">Journal of Number Theory</searchLink>. Sep2026, Vol. 286, p108-130. 23p. – Name: Subject Label: Subjects Group: Su Data: <searchLink fieldCode="DE" term="%22Diophantine+equations%22">Diophantine equations</searchLink><br /><searchLink fieldCode="DE" term="%22Quadratic+fields%22">Quadratic fields</searchLink><br /><searchLink fieldCode="DE" term="%22Modular+forms%22">Modular forms</searchLink><br /><searchLink fieldCode="DE" term="%22Algebraic+number+theory%22">Algebraic number theory</searchLink><br /><searchLink fieldCode="DE" term="%22Mathematical+bounds%22">Mathematical bounds</searchLink><br /><searchLink fieldCode="DE" term="%22Number+theory%22">Number theory</searchLink> – Name: Abstract Label: Abstract Group: Ab Data: In this paper, we investigate solutions to the Diophantine equation A a p + B b p = C c 3 over number fields using the modular method. Assuming certain standard modularity conjectures, we first establish an asymptotic result for general number fields satisfying an appropriate S -unit condition. In particular, we verify that this condition holds for several imaginary quadratic fields. Beyond the asymptotic setting, we also obtain an effective result. Specifically, for the equation a p + d b p = c 3 over K = Q (− d) with d ∈ { 7 , 19 , 43 , 67 } , we determine an explicit bound (depending on d) such that no non-trivial solutions of a certain type exist whenever p exceeds this bound. [ABSTRACT FROM AUTHOR] – Name: AbstractSuppliedCopyright Label: Group: Ab Data: <i>Copyright of Journal of Number Theory is the property of Academic Press Inc. and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.) |
| PLink | https://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=egs&AN=193094113 |
| RecordInfo | BibRecord: BibEntity: Identifiers: – Type: doi Value: 10.1016/j.jnt.2026.02.003 Languages: – Code: eng Text: English PhysicalDescription: Pagination: PageCount: 23 StartPage: 108 Subjects: – SubjectFull: Diophantine equations Type: general – SubjectFull: Quadratic fields Type: general – SubjectFull: Modular forms Type: general – SubjectFull: Algebraic number theory Type: general – SubjectFull: Mathematical bounds Type: general – SubjectFull: Number theory Type: general Titles: – TitleFull: Non-trivial solutions of Aap + Bbp = Cc3 over number fields. Type: main BibRelationships: HasContributorRelationships: – PersonEntity: Name: NameFull: Kara, Yasemin – PersonEntity: Name: NameFull: Nomden, Stef – PersonEntity: Name: NameFull: Özman, Ekin IsPartOfRelationships: – BibEntity: Dates: – D: 01 M: 09 Text: Sep2026 Type: published Y: 2026 Identifiers: – Type: issn-print Value: 0022314X Numbering: – Type: volume Value: 286 Titles: – TitleFull: Journal of Number Theory Type: main |
| ResultId | 1 |
