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The Generalized Fermat Equation

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Open Problems in Mathematics

Abstract

We survey approaches to solving the generalized Fermat equation

$$\displaystyle{x^{p} + y^{q} = z^{r}}$$

in relatively prime integers x, y and z, and integers p, q and r ≥ 2.

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Notes

  1. 1.

    One verifies this by letting t = x ± y and x = t ∓ y, as well as using the fact that (t, y) = 1.

  2. 2.

    The second author found this proof, confronted with the own incapacity to recall the details of the classical proofs, for a seminar. It is possible that the proof may have been known, but we found no reference to it in the literature

  3. 3.

    One would expect, for various reasons, that regular primes occur more frequently than irregular ones. If we knew this, since it has been proved that there are infinitely many irregular primes, Kummer’s result would already imply that there are infinitely many primes p for which FLT holds. However no proof of the fact that the set of regular primes is infinite is known, even now.

References

  1. S. Anni and S. Siksek, On the generalized Fermat equation \(x^{2\ell} + y^{2m} = z^{p}\) for 3 ≤ p ≤ 13, preprint, arXiv 1506.02860.

    Google Scholar 

  2. K. Belabas, F. Beukers, P. Gaudry, H. Lenstra, W. McCallum, B. Poonen, S. Siksek, M. Stoll, M. Watkins, Explicit Methods in Number Theory: Rational Points and Diophantine Equations, Panoramas et synthèses 36, Société Mathématique de France, Paris, 2012.

    MATH  Google Scholar 

  3. M. A. Bennett and I. Chen, Multi-Frey \(\mathbb{Q}\) -curves and the Diophantine equation \(a^{2} + b^{6} = c^{n}\), Algebra Number Theory 6 (2012), 707–730.

    Article  MathSciNet  MATH  Google Scholar 

  4. M. A. Bennett, I. Chen, S. R. Dahmen and S. Yazdani, Generalized Fermat equations: a miscellany, Int. J. Number Theory 11 (2015), 1–28.

    Article  MathSciNet  MATH  Google Scholar 

  5. M. A. Bennett, J. S. Ellenberg and N. Ng, The Diophantine equation \(A^{4} + 2^{\delta }B^{2} = C^{n}\), Int. J. Number Theory 6 (2010), 311–338.

    Article  MathSciNet  MATH  Google Scholar 

  6. C. Breuil, B. Conrad, F. Diamond and R. Taylor, On the modularity of elliptic curves over \(\mathbb{Q}\) : wild 3-adic exercises, J. Amer. Math. Soc. 14 (2001), 843–939.

    Article  MathSciNet  MATH  Google Scholar 

  7. H. Cohen, Number theory. Vol. II. Analytic and modern tools, Graduate Texts in Mathematics, 240, Springer, New York, 2007.

    Google Scholar 

  8. J. Cremona, Elliptic Curve Data, September 2015.

    Google Scholar 

  9. H. Darmon and A. Granville, On the equations z m = F(x,y) and \(Ax^{p} + By^{q} = Cz^{r}\), Bull. London Math. Soc. textbf27 (1995), 513–543.

    Google Scholar 

  10. H. Darmon and L. Merel, Winding quotients and some variants of Fermat’s last theorem, J. Reine Angew. Math. 490 (1997), 81–100.

    MathSciNet  MATH  Google Scholar 

  11. J. Edwards, A complete solution to \(X^{2} + Y ^{3} + Z^{5} = 0\), J. Reine Angew. Math.571 (2004), 213–236.

    MathSciNet  MATH  Google Scholar 

  12. J. S. Ellenberg, Galois representations attached to \(\mathbb{Q}\) -curves and the generalized Fermat equation \(A^{4} + B^{2} = C^{p}\), Amer. J. Math. 126 (2004), 763–787.

    Article  MathSciNet  MATH  Google Scholar 

  13. G. Faltings, Endlichkeitssätze für abelsche Varietäten über Zahlkörpen, Invent. Math. 73 (1983), 349–366.

    Article  MathSciNet  Google Scholar 

  14. N. Freitas, B. Le Hung and S. Siksek, Elliptic curves over real quadratic fields are modular, Invent. Math. 201 (2015), 159–206.

    Article  MathSciNet  MATH  Google Scholar 

  15. N. Freitas and S. Siksek, The asymptotic Fermat’s Last Theorem for five-sixths of real quadratic fields, Compos. Math. 151 (2015), 1395–1415.

    Article  MathSciNet  MATH  Google Scholar 

  16. N. Freitas and S. Siksek, Fermat’s last theorem over some small real quadratic fields, Algebra Number Theory 9 (2015), 875–895.

    Article  MathSciNet  MATH  Google Scholar 

  17. N. Freitas and S. Siksek, Criteria for irreducibility of \(\mod p\) representations of Frey curves, J. Théor. Nombres Bordeaux 27, 2015, 67–76.

    Article  MathSciNet  MATH  Google Scholar 

  18. G. Frey, Links between stable elliptic curves and certain Diophantine equations, Ann. Univ. Sarav. Ser. Math. 1 (1986), iv+40.

    Google Scholar 

  19. Y. Hellegouarch, Points d’ordre 2p h sur les courbes elliptiques, Acta Arith. 26 (1974/75), 253–263.

    Google Scholar 

  20. C. Khare and J.-P. Wintenberger, Serre’s modularity conjecture. I, Invent. Math. 178 (2009), 485–504.

    Article  MathSciNet  MATH  Google Scholar 

  21. C. Khare and J.-P. Wintenberger, Serre’s modularity conjecture. II, Invent. Math. 178 (2009), 505–586.

    Article  MathSciNet  MATH  Google Scholar 

  22. A. Kraus, Majorations effectives pour l’équation de Fermat généralisée, Canad. J. Math. 49 (1997), 1139–1161.

    Article  MathSciNet  Google Scholar 

  23. S. Lang, Cyclotomic fields I and II, Graduate Texts in Mathematics, 121, Springer, New York, 1990.

    Google Scholar 

  24. B. Mazur, Rational isogenies of prime degree, Invent. Math. 44 (1978), 129–162.

    Article  MathSciNet  MATH  Google Scholar 

  25. L. Merel, Bornes pour la torsion des courbes elliptiques sur les corps de nombres, Invent. Math. 124 (1996), 437–449.

    Article  MathSciNet  Google Scholar 

  26. M. Th. Rassias, Problem-solving and selected topics in number theory. In the spirit of the mathematical olympiads. With a foreword by Preda Mihilescu., Springer, NewYork, 2011.

    Google Scholar 

  27. P. Ribenboim, 13 Lectures on Fermat’s Last Theorem, Springer, 1979.

    Book  MATH  Google Scholar 

  28. K. Ribet, On modular representations of \(\text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\) arising from modular forms, Invent. Math. 100 (1990), 431–476.

    Article  MathSciNet  MATH  Google Scholar 

  29. J.-P. Serre, Sur les représentations modulaires de degré 2 de \(\mathrm{Gal}(\overline{ {\mathbf{Q}}/} {\mathbf{Q}})\), Duke Math. J. 54 (1987), 179–230.

    Article  MathSciNet  Google Scholar 

  30. S. Siksek, The modular approach to Diophantine equations, pages 151–179 of [2].

    Google Scholar 

  31. J. H. Silverman, Advanced topics in the arithmetic of elliptic curves, Graduate Texts in Mathematics, 151, Springer-Verlag, New York, 1994.

    Book  Google Scholar 

  32. S. Sitaraman, Vandiver revisited, J. Number Theory 57 (1996), 122–129.

    Article  MathSciNet  MATH  Google Scholar 

  33. R. Taylor and A. Wiles, Ring-theoretic properties of certain Hecke algebras, Ann. of Math. 141 (1995), 553–572.

    Article  MathSciNet  MATH  Google Scholar 

  34. G. Terjanian, Sur l’équation \(x^{2p} + y^{2p} = z^{2p}\), Comptes Rendus Académie de Sciences Paris, 285 (1977), 973–975.

    MathSciNet  MATH  Google Scholar 

  35. F. Thaine, On the ideal class groups of real abelian number fields., Ann. of Math. 128 (1988), 1–18.

    Article  MathSciNet  MATH  Google Scholar 

  36. L. Washington, Introduction to Cyclotomic Fields, Graduate Texts in Mathematics, Springer, New York, 1996.

    Google Scholar 

  37. A. Wiles, Modular elliptic curves and Fermat’s Last Theorem, Ann. Math. 141 (1995), 443–551.

    Article  MathSciNet  MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Mathematics, University of British Columbia, Vancouver, BC, Canada, V6T 1Z2

    Michael Bennett

  2. Mathematisches Institut der Universität Göttingen, Göttingen, Germany

    Preda Mihăilescu

  3. Mathematics Institute, University of Warwick, Coventry, CV4 7AL, UK

    Samir Siksek

Authors
  1. Michael Bennett
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  2. Preda Mihăilescu
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  3. Samir Siksek
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Corresponding author

Correspondence to Preda Mihăilescu .

Editor information

Editors and Affiliations

  1. Department of Mathematics, Senior Research Mathematician, Princeton University, Princeton, New Jersey, USA

    John Forbes Nash, Jr.

  2. Department of Mathematics, Princeton University, Princeton, New Jersey, USA

    Michael Th. Rassias

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Bennett, M., Mihăilescu, P., Siksek, S. (2016). The Generalized Fermat Equation. In: Nash, Jr., J., Rassias, M. (eds) Open Problems in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-319-32162-2_3

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