Skip to main content

A purely exponential Diophantine equation in three unknowns


For any fixed integers a and b greater than 1, we study the Diophantine equation \(a^x+(ab+1)^y=b^z\). First, we describe a heuristic list of the positive integer solutions xy and z of the equation. Finally, we solve the equation in some particular cases, which supports the validity of our list of solutions.

This is a preview of subscription content, access via your institution.


  1. C. Bertók, L. Hajdu, A Hasse-type principle for exponential diophantine equations and its applications. Math. Comput. 85, 849–860 (2016)

    MathSciNet  Article  Google Scholar 

  2. W. Bosma, J. Cannon, C. Playoust, The Magma algebra system i: the user language. J. Symb. Comput. 24, 235–265 (1997)

    MathSciNet  Article  Google Scholar 

  3. Y. Bugeaud, Linear forms in two \(m\)-adic logarithms and applications to Diophantine problems. Compos. Math. 132, 137–158 (2002)

    MathSciNet  Article  Google Scholar 

  4. M. Cipu, M. Mignotte, On a conjecture on exponential Diophantine equations. Acta Arith. 140, 251–269 (2009)

    MathSciNet  Article  Google Scholar 

  5. J.H. Evertse, K. Győry, Unit Equations in Diophantine Number Theory (Cambridge University Press, Cambridge, 2015)

    Book  Google Scholar 

  6. Y. Hu, M. Le, An upper bound for the number of solutions of ternary purely exponential Diophantine equations. J. Number Theory 183, 62–73 (2018)

    MathSciNet  Article  Google Scholar 

  7. Y. Hu, M. Le, An upper bound for the number of solutions of ternary purely exponential Diophantine equations II. Publ. Math. Debr. 95, 335–354 (2019)

    MathSciNet  Article  Google Scholar 

  8. M. Le, R. Scott, R. Styer, A survey on the ternary purely exponential diophantine equation \(a^x+b^y=c^z\). Surv. Math. Appl. 14, 109–140 (2019)

    MathSciNet  MATH  Google Scholar 

  9. F. Luca, On the system of Diophantine equations \(a^2+b^2=(m^2+1)^r\) and \(a^x+b^y=(m^2+1)^z\). Acta Arith. 153, 373–392 (2012)

    MathSciNet  Article  Google Scholar 

  10. T. Miyazaki, Upper bounds for solutions of an exponential Diophantine equation. Rocky Mountain J. Math. 45, 303–344 (2015)

    MathSciNet  Article  Google Scholar 

  11. T. Miyazaki, Contributions to some conjectures on a ternary exponential Diophantine equation. Acta Arith. 186, 1–36 (2018)

    MathSciNet  Article  Google Scholar 

  12. T. Miyazaki, N. Terai, A study on the exponential Diophantine equation \(a^x+(a+b)^y=b^z\). Publ. Math. Debr. 95(1–2), 19–37 (2019)

    Article  Google Scholar 

  13. T. Miyazaki, A. Togbé, P. Yuan, On the Diophantine equation \(a^x + b^y = (a + 2)^z\). Acta Math. Hungar. 149, 1–9 (2016)

    MathSciNet  Article  Google Scholar 

  14. P. Ribenboim, Catalan’s Conjecture: Are 8 and 9 the Only Consecutive Powers? (Academic Press, Boston, 1994)

    MATH  Google Scholar 

  15. R. Scott, On the equations \(p^x-b^y=c\) and \(a^x+b^y=c^z\). J. Number Theory 44, 153–165 (1993)

    MathSciNet  Article  Google Scholar 

  16. R. Scott, R. Styer, Number of solutions to \(a^x+b^y=c^z\). Publ. Math. Debr. 88, 131–138 (2016)

    Article  Google Scholar 

  17. W. Sierpiński, On the equation \(3^x+4^y=5^z\). Wiadom. Mat. 1, 194–195 (1955/56) (in Polish)

  18. N. Terai, On Jeśmanowicz’ conjecture concerning primitive Pythagorean triples. J. Number Theory 141, 316–323 (2014)

    MathSciNet  Article  Google Scholar 

Download references


We would like to thank the referee for the helpful remarks. The first author is supported by JSPS KAKENHI (No. 20K03553). The third author is supported by JSPS KAKENHI (No. 18K03247).

Author information

Authors and Affiliations


Corresponding author

Correspondence to Takafumi Miyazaki.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Miyazaki, T., Sudo, M. & Terai, N. A purely exponential Diophantine equation in three unknowns. Period Math Hung 84, 287–298 (2022).

Download citation

  • Accepted:

  • Published:

  • Issue Date:

  • DOI:


  • S-unit equation
  • Purely exponential equation
  • Baker’s method
  • Simultaneous non-Archimedean valuations

Mathematics Subject Classification

  • 11D61
  • 11J86