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Asymptotic Fermat for signatures ( p, p, 2 ) $(p,p,2)$ and ( p, p, 3 ) $(p,p,3)$ over totally real fields
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Mocanu, Diana (2022) Asymptotic Fermat for signatures ( p, p, 2 ) $(p,p,2)$ and ( p, p, 3 ) $(p,p,3)$ over totally real fields. Mathematika, 68 (4). pp. 1233-1257. doi:10.1112/mtk.12162 ISSN 0025-5793.
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Official URL: https://doi.org/10.1112/mtk.12162
Abstract
Let K be a totally real number field and consider a Fermat‐type equation A a p + B b q = C c r $Aa^p+Bb^q=Cc^r$ over K. We call the triple of exponents ( p , q , r ) $(p,q,r)$ the signature of the equation. We prove various results concerning the solutions to the Fermat equation with signature ( p , p , 2 ) $(p,p,2)$ and ( p , p , 3 ) $(p,p,3)$ using a method involving modularity, level lowering and image of inertia comparison. These generalize and extend the recent work of Işik, Kara and Özman. For example, consider K a totally real field of degree n with 2 ∤ h K + $2 \nmid h_K^+$ and 2 inert. Moreover, suppose there is a prime q ⩾ 5 $q\geqslant 5$ which totally ramifies in K and satisfies gcd ( n , q − 1 ) = 1 $\gcd (n,q-1)=1$ , then we know that the equation a p + b p = c 2 $a^p+b^p=c^2$ has no primitive, non‐trivial solutions ( a , b , c ) ∈ O K 3 $(a,b,c) \in \mathcal {O}_K^3$ with 2 | b $2 | b$ for p sufficiently large.
| Item Type: | Journal Article | ||||||||
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| Subjects: | Q Science > QA Mathematics | ||||||||
| Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | ||||||||
| SWORD Depositor: | Library Publications Router | ||||||||
| Library of Congress Subject Headings (LCSH): | Fermat's theorem, Curves, Elliptic, Formally real fields, Algebraic number theory | ||||||||
| Journal or Publication Title: | Mathematika | ||||||||
| Publisher: | London Mathematical Society | ||||||||
| ISSN: | 0025-5793 | ||||||||
| Official Date: | 2022 | ||||||||
| Dates: |
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| Volume: | 68 | ||||||||
| Number: | 4 | ||||||||
| Page Range: | pp. 1233-1257 | ||||||||
| DOI: | 10.1112/mtk.12162 | ||||||||
| Status: | Peer Reviewed | ||||||||
| Publication Status: | Published | ||||||||
| Reuse Statement (publisher, data, author rights): | ** Article version: VoR ** From Wiley via Jisc Publications Router ** History: received 06-03-2022; rev-recd 08-07-2022; accepted 24-07-2022; pub-electronic 13-09-2022; pub-print 10-2022. ** Licence for VoR version of this article: http://creativecommons.org/licenses/by/4.0/ | ||||||||
| Access rights to Published version: | Open Access (Creative Commons) | ||||||||
| Date of first compliant deposit: | 28 October 2022 | ||||||||
| Date of first compliant Open Access: | 28 October 2022 |
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