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A purely exponential Diophantine equation in three unknowns

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Abstract

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 x, y and z of the equation. Finally, we solve the equation in some particular cases, which supports the validity of our list of solutions.

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References

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  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)

    Article  MathSciNet  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)

    Article  MathSciNet  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)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  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)

    Article  MathSciNet  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)

    Article  MathSciNet  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)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

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).

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Authors and Affiliations

  1. Division of Pure and Applied Science, Faculty of Science and Technology, Gunma University, Tenjin-cho 1-5-1, Kiryu, 376-8515, Japan

    Takafumi Miyazaki

  2. Professor Emeritus, Seikei University, Kichijhoji Kitamachi 3-3-1, Musashino-shi, 180-8633, Japan

    Masaki Sudo

  3. Division of Mathematical Sciences, Department of Integrated Science and Technology, Faculty of Science and Technology, Oita University, 700 Dannoharu, Oita, 870–1192, Japan

    Nobuhiro Terai

Authors
  1. Takafumi Miyazaki
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  2. Masaki Sudo
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  3. Nobuhiro Terai
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Correspondence to Takafumi Miyazaki.

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Miyazaki, T., Sudo, M. & Terai, N. A purely exponential Diophantine equation in three unknowns. Period Math Hung 84, 287–298 (2022). https://doi.org/10.1007/s10998-021-00405-x

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  • DOI: https://doi.org/10.1007/s10998-021-00405-x

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