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An application of Baker’s method to the Jeśmanowicz’ conjecture on primitive Pythagorean triples

  • Maohua Le
  • Gökhan SoydanEmail author
Article

Abstract

Let m, n be positive integers such that \(m>n\), \(\gcd (m,n)=1\) and \(m \not \equiv n \pmod {2}\). In 1956, L. Jeśmanowicz conjectured that the equation \((m^2 - n^2)^x + (2mn)^y = (m^2+n^2)^z\) has only the positive integer solution \((x,y,z) = (2,2,2)\). This conjecture is still unsolved. In this paper, combining a lower bound for linear forms in two logarithms due to M. Laurent with some elementary methods, we prove that if \(mn \equiv 2 \pmod {4}\) and \(m > 30.8 n\), then Jeśmanowicz’ conjecture is true.

Keywords

Ternary purely exponential Diophantine equation Primitive Pythagorean triple Jeśmanowicz’ conjecture Application of Baker’s method 

Mathematics Subject Classification

11D61 11J86 

Notes

Acknowledgements

The authors would like to thank Professors Reese Scott and Robert Styer and the anonymous reviewers for reading the manuscript carefully and giving valuable advices. Especially thanks to Professor Robert Styer for checking the calculations and providing other technical assistance. The second author was supported by the Research Fund of Bursa Uludağ University under project number F-2016/9.

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Copyright information

© Akadémiai Kiadó, Budapest, Hungary 2019

Authors and Affiliations

  1. 1.Institute of MathematicsLingnan Normal CollegeZhangjiangChina
  2. 2.Department of MathematicsBursa Uludağ UniversityBursaTurkey

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