Skip to main content
SpringerLink
Log in

The polynomial–exponential equation \(1+2^a+6^b=y^q\)

  • Published:
Periodica Mathematica Hungarica Aims and scope Submit manuscript

Abstract

We solve the equation of the title for \(q=3\) and, partially, for \(q=2\). These are the only prime values of q for which there exist integer solutions. Our arguments are based upon off-diagonal Padé approximation to the binomial function.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. M. Bauer, M. Bennett, Application of the hypergeometric method to the generalized Ramanujan–Nagell equation. Ramanujan J. 6, 209–270 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  2. M. Bennett, Effective measures of irrationality for certain algebraic numbers. J. Aust. Math. Soc. 62, 329–344 (1997)

    Article  MATH  Google Scholar 

  3. M. Bennett, Y. Bugeaud, M. Mignotte, Perfect powers with few binary digits and related Diophantine problems, II. Math. Proc. Camb. Philos. Soc. 153, 525–540 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  4. A. Berczes, L. Hajdu, T. Miyazaki, I. Pink, On the Diophantine Equation \(1+x^a+z^b=y^n\). J. Comb. Number Theory 8(2), 145–154 (2016)

    MathSciNet  MATH  Google Scholar 

  5. P. Corvaja, U. Zannier, On the Diophantine equation \(f(a^m, y)=b^n\). Acta Arith. 94, 25–40 (2000)

    MathSciNet  MATH  Google Scholar 

  6. P. Corvaja, U. Zannier, \(S\)-unit points on analytic hypersurfaces. Ann. Sci. École Norm. Sup. 38, 76–92 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  7. P. Corvaja, U. Zannier, Applications of the Subspace Theorem to Certain Diophantine Problems: A survey of Some Recent Results. Diophantine Approximation, Developments in Mathematics, vol 16 (Springer, Wien NewYork Vienna, 2008), pp. 161–174

  8. L. Hajdu, I. Pink, On the Diophantine equation \(1+2^a+x^b=y^n\). J. Number Theory 143, 1–13 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  9. F. Luca, On the equation \(1!^k+2!^k + \cdots + n!^k = x^2\). Period. Math. Hung. 44, 219–224 (2002)

    Article  MATH  Google Scholar 

  10. K. Mahler, On the fractional parts of the powers of a rational number (II). Mathematika 4, 122–124 (1957)

    Article  MathSciNet  MATH  Google Scholar 

  11. D. Ridout, Rational approximations to algebraic numbers. Mathematika 4, 125–131 (1957)

    Article  MathSciNet  MATH  Google Scholar 

  12. U. Zannier, Diophantine equations with linear recurrences. An overview of some recent progress. J. Théor. Nr. Bordx. 17, 423–435 (2005)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

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

    Michael A. Bennett

Authors
  1. Michael A. Bennett
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Michael A. Bennett.

Additional information

Michael A. Bennett was supported by NSERC.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bennett, M.A. The polynomial–exponential equation \(1+2^a+6^b=y^q\) . Period Math Hung 75, 387–397 (2017). https://doi.org/10.1007/s10998-017-0208-x

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10998-017-0208-x

Keywords

Mathematics Subject Classification

Search

Navigation