Skip to main content
SpringerLink
Log in

An Equation of Goormaghtigh and Diophantine Approximations

  • Chapter
Current Trends in Number Theory

Abstract

An equation of Goormaghtigh (1917) states

$$\frac{{{x^m} - 1}} {{x - 1}} = \frac{{{y^n} - 1}} {{y - 1}}\quad {\text{in}}\,{\text{integers}}\quad {\text{x}} > {\text{1}},{\text{y}} > {\text{1}},{\text{m}} > {\text{2}},{\text{n}} > {\text{2}}.$$
((1))

We give an account of recent results on (1) and we refer to [26] for a survey. In fact the present article can be viewed as updating Section 3 of [26]. Further, we shall consider an extension of (1) with m = 3 and derive a new result from a recent theorem of Bilu, Hanrot and Voutier [4] on primitive divisors of Lucas and Lehmer sequences. We shall also discuss some general results on diophantine approximations by applying them to (1). All the constants appearing in this article are effectively computable. This means that they can be determined explicitly in terms of various parameters involved. By C = C(ϰ), we understand that C is a number depending only on Κ.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 40.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. A. Baker, Simultaneous rational approximations to certain algebraic numbers. Proc. Cambridge Phil. Soc. 63 (1967), 693–702.

    Article  MathSciNet  MATH  Google Scholar 

  2. R. Balasubramanian and T.N. Shorey, On the equation a(xm− 1)/(x − 1) = b(yn − 1)/(y − 1), Math. Scand. 46 (1980), 177–182.

    Article  MathSciNet  MATH  Google Scholar 

  3. M. Bennett, Rational approximation to algebraic numbers of small height: The diophantine equation |axn − byn| = 1, J. reine angew. Math., to appear.

    Google Scholar 

  4. Y.F. Bilu, G. Hanrot and P.M. Voutier, Existence of primitive divisors of Lucas and Lehmer numbers, J. reine angew. Math., to appear.

    Google Scholar 

  5. Y. Bugeaud and T.N. Shorey, On the number of solutions of the generalized Ramanujan-Nagell equation, J. reine angew. Math., to appear.

    Google Scholar 

  6. Y. Bugeaud and T.N. Shorey, On the equation (xm− 1)/(x − 1) = (yn− 1)/(y − 1), Pacific J. Math., to appear.

    Google Scholar 

  7. J. Coates, An effective p-adic analogue of a theorem of Thue, Acta Arith. 15 (1969), 279–305.

    MathSciNet  MATH  Google Scholar 

  8. H. Davenport, D.J. Lewis and A. Schinzel, Equations of the form f(x) = g(y), Quart. J. Math. Oxford (2) 12 (1961), 304–312.

    Article  MathSciNet  MATH  Google Scholar 

  9. M. van Frankenhuysen, A lower bound in the abc-conjecture, J. Number Theory 82 (2000), 91–95.

    Article  MathSciNet  MATH  Google Scholar 

  10. R. Goormaghtigh, L’Intermédiaire des Mathématiciens 24 (1917), 88.

    Google Scholar 

  11. J.H. Loxton, Some problems involving powers of integers, Acta Arith. 46 (1986), 113–123.

    MathSciNet  MATH  Google Scholar 

  12. Maohua Le, On the diophantine equation (x3 − 1)/(x − 1) = (yn− 1)/(y − 1). Trans. Amer. Math. Soc. 351 (1999), 1063–1074.

    Article  MathSciNet  MATH  Google Scholar 

  13. Maohua Le, Exceptional solutions of the exponential diophantine equation (x3 − 1)/(x − 1) = (yn− 1)/(y − 1), J. reine angew. Math., to appear.

    Google Scholar 

  14. M. Mignotte, A Corollary to a theorem of Laurent-Mignotte-Nesterenko, Acta Arith. 86 (1998), 215–225.

    MathSciNet  MATH  Google Scholar 

  15. T. Nagell, The diophantine eguation x2+ 7 = 2n, Ark. Mat. 4 (1961), 185–188.

    Article  MathSciNet  Google Scholar 

  16. Yu. V. Nesterenko and T.N. Shorey, On an equation of Goormaghtigh, Acta Arith. 83 (1998), 381–389.

    MathSciNet  MATH  Google Scholar 

  17. K. Ratat, L’Intermédiaire des Mathématiciens 23 (1916), 150.

    Google Scholar 

  18. C. Runge, Ueber ganzzahlige Lösungen von Gleichungen zwischen swei Veränderlichen, J. reine angew. Math. 100 (1887), 425–435.

    MathSciNet  MATH  Google Scholar 

  19. T.N. Shorey, Linear forms in the logarithms of algebraic numbers with small coefficients I, J. Indian Math. Soc. (N.S.) 38 (1974), 271–284.

    MathSciNet  MATH  Google Scholar 

  20. T.N. Shorey, On gaps between numbers with a large prime factor II, Acta Arith. 25 (1974), 365–373.

    MathSciNet  MATH  Google Scholar 

  21. T.N. Shorey, Perfect powers in values of certain polynomials at integer points, Math. Proc. Camb. Phil. Soc. 99 (1986), 195–207.

    Article  MathSciNet  MATH  Google Scholar 

  22. T.N. Shorey, On the equation axm− byn = k, Indag. Math. 48 (1986), 353–358.

    Article  MathSciNet  MATH  Google Scholar 

  23. T. N. Shorey, Perfect powers in products of integers from a block of consecutive integers, Acta Arith. 49 (1987), 71–79.

    MathSciNet  MATH  Google Scholar 

  24. T.N. Shorey, Some exponential Diophantine equations II, Number Theory and Related Topics ed. by S. Raghavan, Tata Institute of Fundamental Research, Bombay (1988), 217–229.

    Google Scholar 

  25. T.N. Shorey, Integers with identical digits, Acta Arith. 53 (1989), 81–99.

    MathSciNet  MATH  Google Scholar 

  26. T.N. Shorey, Exponential diophantine equations involving products of consecutive integers and related equations, Number Theory edited by R.P. Bambah, V.C. Dumir and R.J. Hans-Gill, Hindustan Book Agency (1999), 463–495.

    Google Scholar 

  27. T.N. Shorey, Mathematical Contributions, Bulletin Bombay Mathematical Colloquium 15 (1999), 1–19.

    MathSciNet  Google Scholar 

  28. T.N. Shorey and Yu. V. Nesterenko, Perfect powers in products of integers from a block of consecutive integers (II), Acta Arith. 76 (1996), 191–198.

    MathSciNet  MATH  Google Scholar 

  29. C.L. Siegel, Approximation algebraischer Zahlen, Math. Z. 10 (1921), 173–213.

    Google Scholar 

  30. C.L. Siegel, Ueber einige Anwendungen diophantischer Approximationen, Abh. Preuss. Akad. Wiss. Phys- math. K1, (1929), No.1, 70 pp.

    Google Scholar 

  31. V.G. Sprindžuk, Effective estimates in ‘ternary’ exponential diophantine equations (Russian), Dokl. Akad. Nauk BSSR 13 (1969), 777–780.

    MathSciNet  Google Scholar 

  32. C.L. Stewart and Kunrui Yu, On the abc-conjecture II, Duke Math. J. to appear.

    Google Scholar 

Download references

Authors
  1. T. N. Shorey
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Harish-Chandra Research Institute, Chhatnag Road, Jhusi, 211 019, Allahabad, India

    Sukumar Das Adhikari  & B. Ramakrishnan  & 

  2. Department of Mathematics, University of Pune, 411 007, Pune, India

    Shashikant A. Katre

Rights and permissions

Reprints and permissions

Copyright information

© 2002 Hindustan Book Agency

About this chapter

Cite this chapter

Shorey, T.N. (2002). An Equation of Goormaghtigh and Diophantine Approximations. In: Adhikari, S.D., Katre, S.A., Ramakrishnan, B. (eds) Current Trends in Number Theory. Hindustan Book Agency, Gurgaon. https://doi.org/10.1007/978-93-86279-09-5_19

Download citation

Publish with us

Policies and ethics

Search

Navigation