Bibliographic Details
| 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] |
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| Database: |
Engineering Source |
