Advertisement

The Ramanujan Journal

, Volume 6, Issue 2, pp 209–270 | Cite as

Applications of the Hypergeometric Method to the Generalized Ramanujan-Nagell Equation

  • Mark Bauer
  • Michael A. Bennett
Article

Abstract

In this paper, we refine work of Beukers, applying results from the theory of Padé approximation to (1 − z)1/2 to the problem of restricted rational approximation to quadratic irrationals. As a result, we derive effective lower bounds for rational approximation to \(\sqrt m\) (where m is a positive nonsquare integer) by rationals of certain types. Forexample, we have
$$\left| {\sqrt 2 - \frac{p}{q}} \right| \gg q^{ - 1.47} {\text{ and }}\left| {\sqrt 3 - \frac{p}{q}} \right| \gg q^{ - 1.62}$$
provided q is a power of 2 or 3, respectively. We then use this approach to obtain sharp bounds for the number of solutions to certain families of polynomial-exponential Diophantine equations. In particular, we answer a question of Beukers on the maximal number of solutions of the equation x2 + D = pn where D is a nonzero integer and p is an odd rational prime, coprime to D.
Ramanujan-Nagell equation hypergeometric method Padé approximation 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    R. Apéry, “Sur une équation diophantienne,” C. R. Acad. Sci. Paris Sér. A 251 (1960), 1451-1452.Google Scholar
  2. 2.
    A. Baker, “Rational approximations to certain algebraic numbers,” Proc. London Math. Soc. 14(3) (1964), 385-398.Google Scholar
  3. 3.
    A. Baker, “Rational approximations to 3?2 and other algebraic numbers,” Quart. J. Math. Oxford Ser. 15(2) (1964), 375-383.Google Scholar
  4. 4.
    E. Bender and N. Herzberg, “Some Diophantine equations related to the quadratic form ax 2 + by 2,” Bull. Amer. Math. Soc. 81 (1975), 161-162.Google Scholar
  5. 5.
    E. Bender and N. Herzberg, “Some Diophantine equations related to the quadratic form ax 2 + by 2,” Studies in algebra and number theory, Adv. in Math. Suppl. Stud., Vol. 6, Academic Press, New York-London, 1979, pp. 219-272.Google Scholar
  6. 6.
    M. Bennett, “Simultaneous rational approximation to binomial functions,” Trans. Amer. Math. Soc. 348 (1996), 1717-1738.Google Scholar
  7. 7.
    M. Bennett, “Effective measures of irrationality for certain algebraic numbers,” J. Austral. Math. Soc. 62 (1997), 329-344.Google Scholar
  8. 8.
    M. Bennett, “Explicit lower bounds for rational approximation to algebraic numbers,” Proc. London Math. Soc. 75 (1997), 63-78.Google Scholar
  9. 9.
    F. Beukers, The generalised Ramanujan-Nagell equation. Dissertation, Rijksuniversiteit, Leiden, 1979. With a Dutch summary. Rijksuniversiteit te Leiden, Leiden, 1979, 57 pp.Google Scholar
  10. 10.
    F. Beukers, “On the generalized Ramanujan-Nagell equation I,” Acta Arith. 38 (1980/1981), 389-410.Google Scholar
  11. 11.
    F. Beukers, “On the generalized Ramanujan-Nagell equation II,” Acta Arith. 39 (1981), 113-123.Google Scholar
  12. 12.
    Y. Bilu, G. Hanrot, and P. Voutier, “Existence of primitive divisors of Lucas and Lehmer numbers,” J. Reine Angew. Math. 539 (2001), 75-122.Google Scholar
  13. 13.
    Y. Bugeaud and T. Shorey, “On the number of solutions of the generalized Ramanujan-Nagell equation,” J. Reine Angew. Math. 539 (2001), 55-74.Google Scholar
  14. 14.
    X. Chen, Y. Guo, and M. Le, “On the number of solutions of the generalized Ramanujan-Nagell equation x 2 + D = k n,” Acta Math. Sinica 41 (1998), 1249-1254.Google Scholar
  15. 15.
    X. Chen and M. Le, “On the number of solutions of the generalized Ramanujan-Nagell equation x 2?D = k n,” Publ. Math. Debrecen 49 (1996), 85-92.Google Scholar
  16. 16.
    G.V. Chudnovsky, “On the method of Thue-Siegel,” Ann. Math. II Ser. 117 (1983), 325-382.Google Scholar
  17. 17.
    E.L. Cohen, “On the Ramanujan-Nagell equation and its generalizations,” Number theory (Banff, AB, 1988), de Gruyter, Berlin, 1990, pp. 81-92.Google Scholar
  18. 18.
    C. Heuberger and M. Le, “On the generalized Ramanujan-Nagell equation x 2 + D = p z,” J. Number Theory 78 (1999), 312-331.Google Scholar
  19. 19.
    M. Le, “On the generalized Ramanujan-Nagell equation x 2 ? D = p n,” Acta Arith. 58 (1991), 289-298.Google Scholar
  20. 20.
    M. Le, “On the number of solutions to the Diophantine equation x 2 ? D = p nActa Math. Sinica 34 (1991), 378-387 (Chinese).Google Scholar
  21. 21.
    M. Le, “On the number of solutions of the generalized Ramanujan-Nagell equation x 2 ? D = 2n+2,” Acta Arith. 60 (1991), 149-167.Google Scholar
  22. 22.
    M. Le, “On the Diophantine equation x 2 + D = 4p n,” J. Number Theory 41 (1992), 87-97.Google Scholar
  23. 23.
    M. Le, “On the generalized Ramanujan-Nagell equation x 2?D = 2n+2,” Trans. Amer.Math. Soc. 334 (1992), 809-825.Google Scholar
  24. 24.
    M. Le, “On the Diophantine equation x 2 ? D = 4p n,” J. Number Theory 41 (1992), 257-271.Google Scholar
  25. 25.
    M. Le, “On the Diophantine equations d 1 x 2 + 22m d 2 = y n and d 1 x 2 + d 2 = 4y n,” Proc. Amer. Math. Soc. 118 (1993), 67-70.Google Scholar
  26. 26.
    M. Le, “On the Diophantine equation D 1 x 2 + D 2 = 2n+2,” Acta Arith. 64 (1993), 29-41.Google Scholar
  27. 27.
    M. Le, “A note on the Diophantine equation x 2 + 4D = y p,” Monatsh. Math. 116 (1993), 283-285.Google Scholar
  28. 28.
    M. Le, “On the number of solutions of the Diophantine equation x 2 + D = p n,” C. R. Acad. Sci. Paris Sér. I Math. 317 (1993), 135-138.Google Scholar
  29. 29.
    M. Le, “On the number of solutions of the generalized Ramanujan-Nagell equation x 2 ? D = p n,” Publ. Math. Debrecen 45 (1994), 239-254.Google Scholar
  30. 30.
    M. Le, “A note on the generalized Ramanujan-Nagell equation,” J. Number Theory 50 (1995), 193-201.Google Scholar
  31. 31.
    M. Le, “A note on the number of solutions of the generalized Ramanujan-Nagell equation x 2 ? D = k n,” Acta Arith. 78 (1996), 11-18.Google Scholar
  32. 32.
    M. Le, “A note on the Diophantine equation D 1 x 2 + D 2 = 2y n,” Publ. Math. Debrecen 51 (1997), 191-198.Google Scholar
  33. 33.
    M. Le, “On the Diophantine equation (x 3 ? 1)/(x ? 1) = (y n ? 1)/(y ? 1),” Trans. Amer. Math. Soc. 351 (1999), 1063-1074.Google Scholar
  34. 34.
    K. Mahler, “Ein Beweis des Thue-Siegelschen Satzes über die Approximation algebraischer Zahlen für binomische Gleichungen,” Math. Ann. 105 (1931), 267-276.Google Scholar
  35. 35.
    K. Mahler, “Zur Approximation algebraischer Zahlen, I: Ueber den grössten Primteiler binärer Formen,” Math. Ann. 107 (1933), 691-730.Google Scholar
  36. 36.
    M. Mignotte, “A corollary to a theorem of Laurent-Mignotte-Nesterenko,” Acta Arith. 86 (1998), 101-111.Google Scholar
  37. 37.
    T. Nagell, “The diophantine equation x 2 + 7 = 2n,” Ark. Math. 4 (1960), 185-187.Google Scholar
  38. 38.
    S. Ramanujan, “Question 464,” J. Indian Math. Soc. 5 (1913), 120.Google Scholar
  39. 39.
    D. Ridout, “The p-adic generalization of the Thue-Siegel-Roth theorem,” Mathematika 5 (1958), 40-48.Google Scholar
  40. 40.
    J.B. Rosser and L. Schoenfeld, “Approximate formulas for some functions of prime numbers,” Ill, J. Math. 6 (1962), 64-94.Google Scholar
  41. 41.
    L. Schoenfeld, “Sharper bounds for the Chebyshev functions ?(x) and ?(x) II,” Math. Comp. 30 (1976), 337-360.Google Scholar
  42. 42.
    C.L. Siegel, “Die Gleichung ax n ? by n = c,” Math. Ann. 114 (1937), 57-68.Google Scholar
  43. 43.
    A. Thue, “Berechnung aller Lösungen gewisser Gleichungen von der form,” Vid. Skrifter I Mat.-Naturv. Klasse (1918), 1-9.Google Scholar
  44. 44.
    N. Tzanakis and J. Wolfskill, “On the Diophantine equation y 2 = 4q n +4q+1,” J. Number Theory 23 (1986), 219-237.Google Scholar
  45. 45.
    N. Tzanakis and J.Wolfskill, “The Diophantine equation x 2 = 4q a/2 +4q +1, with an application to coding theory,” J. Number Theory 26 (1987), 96-116.Google Scholar
  46. 46.
    T. Xu and M. Le, “On the Diophantine equation D 1 x 2+D 2 = k n,” Publ. Math. Debrecen 47 (1995), 293-297.Google Scholar
  47. 47.
    P. Yuan, “On the number of the solutions of x 2D = p n,” Sichuan Daxue Xuebao 35 (1998), 311-316Google Scholar

Copyright information

© Kluwer Academic Publishers 2002

Authors and Affiliations

  • Mark Bauer
    • 1
  • Michael A. Bennett
    • 1
  1. 1.Department of MathematicsUniversity of IllinoisUrbana

Personalised recommendations