Solving Fermat-type equations 
Authors:
Nicolas Billerey and Luis V. Dieulefait
Journal:
Math. Comp. 79 (2010), 535-544
MSC (2000):
Primary 11F11, 11D41, 14H52; Secondary 11D59
DOI:
https://doi.org/10.1090/S0025-5718-09-02294-7
Published electronically:
July 22, 2009
MathSciNet review:
2552239
Full-text PDF
Abstract | References | Similar Articles | Additional Information
Abstract: In this paper, we are interested in solving the Fermat-type equations 
, where 
is a positive integer and 
a prime number 
. We describe a new method based on modularity theorems which allows us to improve all earlier results for this equation. We finally discuss the present limits of the method by looking at the case 
.
- 1. Nicolas Billerey, Équations de Fermat de type (5,5,𝑝), Bull. Austral. Math. Soc. 76 (2007), no. 2, 161–194 (French, with French summary). MR 2353205, https://doi.org/10.1017/S0004972700039575
- 2. J. E. Cremona, Algorithms for modular elliptic curves, 2nd ed., Cambridge University Press, Cambridge, 1997. MR 1628193
- 3. A. Kraus and J. Oesterlé, Sur une question de B. Mazur, Math. Ann. 293 (1992), no. 2, 259–275 (French). MR 1166121, https://doi.org/10.1007/BF01444715
- 4.
W. Stein.
The Modular Forms Database.
http://modular.fas.harvard.edu/Tables, 2004. - 5. N. Tzanakis and B. M. M. de Weger, How to explicitly solve a Thue-Mahler equation, Compositio Math. 84 (1992), no. 3, 223–288. MR 1189890
. Retrieve articles in Mathematics of Computation with MSC (2000): 11F11, 11D41, 14H52, 11D59
Retrieve articles in all journals with MSC (2000): 11F11, 11D41, 14H52, 11D59
Additional Information
Nicolas Billerey
Affiliation:
Université Pierre et Marie Curie – Paris 6, UMR 7586, Case 247, 4, place Jussieu, Institut de Mathématiques, 75252 Paris, France
Email:
billerey@math.jussieu.fr
Luis V. Dieulefait
Affiliation:
Departament d’Algebra i Geometria, Universitat de Barcelona, Gran Via de les Corts Catalanes 585, (08007) Barcelona, Spain
Email:
ldieulefait@ub.edu
DOI:
https://doi.org/10.1090/S0025-5718-09-02294-7
Keywords:
Modular forms,
Fermat's equation,
elliptic curves,
Thue-Mahler equations
Received by editor(s):
July 10, 2008
Received by editor(s) in revised form:
January 28, 2009
Published electronically:
July 22, 2009
Article copyright:
© Copyright 2009
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
