Advertisement

Class Number Conditions for the Diagonal Case of the Equation of Nagell and Ljunggren

  • Preda Mihăilescu
Part of the Developments in Mathematics book series (DEVM, volume 16)

Abstract

The diagonal case of the Nagell-Ljunggren equation is
$$ \frac{{x^p - 1}} {{x - 1}} = p^e \cdot y^p with x,y \in \mathbb{Z} e \in \left\{ {0,1} \right\}, $$
(1)
and p an odd prime. The only known nontrivial solution is
$$ \frac{{18^3 - 1}} {{18 - 1}} = 7^3 , $$
(2)
and it is conjectured to be also the only such solution. However, it is not even proved that (1) has only finitely many solution.

Keywords

Exponential diophantine equation diagonal case of Nagell and Ljunggren 

2000 Mathematics subject classification

11D61 11D45 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [BCEMS]
    Buhler, J., Crandall, R., Emvall, R., Metsänkylä, T., Shokrollahi, A.: Irregular primes and cyclotomic invariants to 12 million. J. Symb. Comput. 31, 89–96 (2001)zbMATHCrossRefGoogle Scholar
  2. [Bl]
    Blatter, C.: Analysis III. Heidelberger Taschenbücher, vol. 153. Springer, Heidelberg (1974)Google Scholar
  3. [BH]
    Bugeaud, Y., Hanrot, G.: Un nouveau critère pour l’équation de Catalan. Matematika 47, 63–73 (2000)zbMATHMathSciNetGoogle Scholar
  4. [BHM]
    Bugeaud, Y., Hanrot, G., Mignotte, M.: Sur l’équation diophantienne \( \frac{{x^{n - 1} }} {{x - 1}} = y^q \), III. Proc. Lond. Math. Soc. 84, 59–78 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  5. [Ca]
    Cassels, J.W.S.: On the equation a x-b y = 1, II. Proc. Camb. Philos. Soc. 56, 97–103 (1960)CrossRefMathSciNetGoogle Scholar
  6. [Ei]
    Eichler, M.: Eine Bemerkung zur Fermatschen Vermutung. Acta Arith. 11, 129–131 (1965) err. 261MathSciNetGoogle Scholar
  7. [Go]
    Gouvêa, F.Q.: p-adic Numbers. An Introduction, 2nd edn. Universitext. Springer, Heidelberg (1991)Google Scholar
  8. [Hy]
    Hyyrö, S.: Über das Catalan’sche Problem. Ann. Univ. Turku Ser. AI 79, 3–10 (1964)Google Scholar
  9. [IR]
    Ireland, K., Rosen, M.: A Classical Introduction to Modern Number Theory, 2nd edn. Graduate Texts in Mathematics, vol. 84. Springer, Heidelberg (1990)Google Scholar
  10. [Iw]
    Iwasawa, K.: A note on Jacobi Sums. In: Informatica Teorica, Strutture in Corpi Algebrici. Istituto Nazionale di Alta Mathematica., Symp. Math., vol. 15, pp. 447–459. Academic Press, London (1975)Google Scholar
  11. [Jh]
    Jha, V.: The Stickelberger Ideal in the Spirit of Kummer with Applications to the First Case of Fermat’s Last Theorem. Queen’s Papers in Pure and Applied Mathematics, vol. 93. Queen’s University, Kingston, Ont. (1993)Google Scholar
  12. [La]
    Lang, S.: Algebraic Number Theory, 2nd edn. Graduate Texts in Mathematics, vol. 110. Springer, Heidelberg (1986)Google Scholar
  13. [La1]
    Lang, S.: Cyclotomic Fields, I and II, combined 2nd edn. Graduate Texts in Mathematics, vol. 121. Springer, Heidelberg (1990)Google Scholar
  14. [Mi]
    Mignotte, M.: Catalan’s equation just before 2000. In: Jutila, M., Metsänkylä, T. (eds.) Number Theory: Proceedings of the Turku Symposium on Number Theory in Memory of Kustaa Inkeri, May 31–June 4, 1999, pp. 247–254. de Gruyter, Berlin (2001)Google Scholar
  15. [Mih]
    Mihăilescu, P.: A class number free criterion for Catalan’s conjecture. J. Number Theory 99, 225–231 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  16. [Mih1]
    Mihăilescu, P.: On the class groups of cyclotomic extensions in presence of a solution to Catalan’s equation. J. Number Theory 118, 123–144 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  17. [Mih2]
    Mihăilescu, P.: Primary cyclotomic units and a proof of Catalan’s conjecture. J. Reine Angew. Math. 572, 167–195 (2004)zbMATHMathSciNetGoogle Scholar
  18. [Ri]
    Ribenboim, P.: Catalan’s Conjecture. Academic Press, London (1994)Google Scholar
  19. [Ri1]
    Ribenboim, P.: 13 Lectures on Fermát’s Last Theorem. Springer, Heidelberg(1979)Google Scholar
  20. [Wa]
    Washington, L.: Introduction to Cyclotomic Fields, 2nd edn. Graduate Texts in Mathematics, vol. 83. Springer, Heidelberg (1996) 273Google Scholar

Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  • Preda Mihăilescu
    • 1
  1. 1.Mathematisches InstitutUniversität GöttingenGöttingenGermany

Personalised recommendations