Diophantine equations after Fermat’s last theorem
Samir Siksek[1]
- [1] Mathematics Institute University of Warwick Coventry, CV4 7AL, United Kingdom
Journal de Théorie des Nombres de Bordeaux (2009)
- Volume: 21, Issue: 2, page 423-434
- ISSN: 1246-7405
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top- C. Batut, K. Belabas, D. Bernardi, H. Cohen and M. Olivier, User’s guide to PARI-GP, version 2.3.2. (See also http://pari.math.u-bordeaux.fr/)
- M. A. Bennett, Powers in recurrence sequences : Pell equations. Trans. Amer. Math. Soc. 357 (2005), 1675–1691. Zbl1125.11019MR2115381
- N. Bruin, On powers as sums of two cubes. Pages 169–184 of Algorithmic number theory (edited by W. Bosma), Lecture Notes in Comput. Sci. 1838, Springer, Berlin, 2000. Zbl0986.11021MR1850605
- W. Bosma, J. Cannon and C. Playoust, The Magma Algebra System I: The User Language. J. Symb. Comp. 24 (1997), 235–265. (See also http://magma.maths.usyd.edu.au/magma/) Zbl0898.68039MR1484478
- C. Breuil, B. Conrad, F. Diamond and R. Taylor, On the modularity of elliptic curves over : wild -adic exercises. J. Amer. Math. Soc. 14 No.4 (2001), 843–939. Zbl0982.11033MR1839918
- Y. Bugeaud, M. Mignotte and S. Siksek, Classical and modular approaches to exponential Diophantine equations I. Fibonacci and Lucas perfect powers. Annals of Mathematics 163 (2006), 969–1018. Zbl1113.11021MR2215137
- Y. Bugeaud, M. Mignotte and S. Siksek, Classical and modular approaches to exponential Diophantine equations II. The Lebesgue–Nagell Equation. Compositio Mathematica 142 (2006), 31–62. Zbl1128.11013MR2196761
- Y. Bugeaud, M. Mignotte and S. Siksek, A multi-Frey approach to some multi-parameter families of Diophantine equations. Canadian Journal of Mathematics 60 (2008), no. 3, 491–519. Zbl1156.11014MR2414954
- I. Chen and S. Siksek, Perfect powers expressible as sums of two cubes. Journal of Algebra, to appear. Zbl1215.11026
- J. H. E. Cohn, On square Fibonacci numbers; J. London Math. Soc. 39 (1964), 537–540. Zbl0127.26705MR163867
- J. H. E. Cohn, The Diophantine equation . Acta Arith. LXV.4 (1993), 367–381. Zbl0795.11016MR1259344
- J. E. Cremona, Algorithms for Modular Elliptic Curves. 2nd edition, Cambridge University Press, 1997. Zbl0872.14041MR1628193
- H. Cohen, Number Theory, Vol. I: Tools and Diophantine Equations and Vol. II: Analytic and Modern Tools. Springer-Verlag, GTM 239, 240, 2007. Zbl1119.11001MR2312337
- S. R. Dahmen, Classical and modular methods applied to Diophantine equations. University of Utrecht Ph.D. thesis, 2008.
- A. Kraus, Sur l’équation . Experimental Mathematics 7 (1998), No. 1, 1–13. Zbl0923.11054MR1618290
- B. Poonen, E. F. Schaefer and M. Stoll, Twists of and primitive solutions to . Duke Math. J. 137 (2007), 103–158. Zbl1124.11019MR2309145
- K. Ribet, On modular representations of arising from modular forms. Invent. Math. 100 (1990), 431–476. Zbl0773.11039MR1047143
- S. Siksek and J. E. Cremona, On the Diophantine equation . Acta Arith. 109.2 (2003), 143–149. Zbl1026.11043MR1980642
- W. A. Stein, Modular Forms: A Computational Approach. American Mathematical Society, Graduate Studies in Mathematics 79, 2007. Zbl1110.11015MR2289048
- R. L. Taylor and A. Wiles, Ring theoretic properties of certain Hecke algebras. Annals of Math. 141 (1995), 553–572. Zbl0823.11030MR1333036
- A. Wiles, Modular elliptic curves and Fermat’s Last Theorem. Annals of Math. 141 (1995), 443–551. Zbl0823.11029MR1333035