Vol. 9, No. 4, 2015

Download this article
Download this article For screen
For printing
Recent Issues

Volume 14
Issue 9, 2295–2574
Issue 8, 2001–2294
Issue 7, 1669–1999
Issue 6, 1331–1667
Issue 5, 1055–1329
Issue 4, 815–1054
Issue 3, 545–813
Issue 2, 275–544
Issue 1, 1–274

Volume 13, 10 issues

Volume 12, 10 issues

Volume 11, 10 issues

Volume 10, 10 issues

Volume 9, 10 issues

Volume 8, 10 issues

Volume 7, 10 issues

Volume 6, 8 issues

Volume 5, 8 issues

Volume 4, 8 issues

Volume 3, 8 issues

Volume 2, 8 issues

Volume 1, 4 issues

The Journal
About the Journal
Editorial Board
Editors’ Interests
Subscriptions
 
Submission Guidelines
Submission Form
Policies for Authors
Ethics Statement
 
ISSN: 1944-7833 (e-only)
ISSN: 1937-0652 (print)
Author Index
To Appear
 
Other MSP Journals
Fermat's last theorem over some small real quadratic fields

Nuno Freitas and Samir Siksek

Vol. 9 (2015), No. 4, 875–895
Abstract

Using modularity, level lowering, and explicit computations with Hilbert modular forms, Galois representations, and ray class groups, we show that for 3 d 23, where d5,17 and is squarefree, the Fermat equation xn + yn = zn has no nontrivial solutions over the quadratic field (d) for n 4. Furthermore, we show that for d = 17, the same holds for prime exponents n 3,5(mod8).

Keywords
Fermat, modularity, Galois representation, level lowering
Mathematical Subject Classification 2010
Primary: 11D41
Secondary: 11F80, 11F03
Milestones
Received: 16 July 2014
Revised: 23 September 2014
Accepted: 9 March 2015
Published: 30 May 2015
Authors
Nuno Freitas
Mathematisches Institut
Universität Bayreuth
95440 Bayreuth
Germany
Samir Siksek
Mathematics Institute
University of Warwick
Coventry
CV4 7AL
United Kingdom