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

  • Wang Tingting
  • Wang Xiaonan
  • Jiang Yingzhao
Original Paper
  • 109 Downloads

Abstract

Let n be a positive integer, and let (abc) be a primitive Pythagorean triple with \(a^2+b^2=c^2\). A positive integer solution (xyz) of the equation \((an)^x+(bn)^y=(cn)^z\) is called exceptional if \((x,y,z)\ne (2,2,2)\). Sixty years ago, L. Jeśmanowicz conjectured that, for any n, the equation has no exceptional solutions. This problem is not resolved as yet. In this paper, using the Baker method, we prove that if \(n>1\), \(b+1=c\) and \(c>500000\), then the equation has no exceptional solutions (xyz) with \(y>z>x\).

Keywords

Exponential diophantine equation Pythagorean triple Jeśmanowicz conjecture Baker method 

Mathematics Subject Classification

11D61 

Notes

Acknowledgements

The authors express their gratitude to the referees for very helpful and detailed comments.

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Copyright information

© Springer-Verlag Italia 2017

Authors and Affiliations

  1. 1.College of ScienceNorthwest A&F UniversityYanglingPeople’s Republic of China

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