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Periodica Mathematica Hungarica

, Volume 75, Issue 2, pp 190–195 | Cite as

On the simultaneous Diophantine equations \( m \cdot (x_1^k+x_2^k+ \cdots + x_{t_1}^k)=n \cdot (y_1^k+y_2^k+ \cdots y_{t_2}^k)\); \(k=1,3\)

  • Farzali IzadiEmail author
  • Mehdi Baghalaghdam
Article
  • 103 Downloads

Abstract

In this paper, we solve the simultaneous Diophantine equations \(m \cdot ( x_{1}^k+ x_{2}^k +\cdots + x_{t_1}^k)=n \cdot (y_{1}^k+ y_{2}^k +\cdots + y_{t_2}^k )\), \(k=1,3\), where \( t_1, t_2\ge 3\), and m, n are fixed arbitrary and relatively prime positive integers. This is done by choosing two appropriate trivial parametric solutions and obtaining infinitely many nontrivial parametric solutions. Also we work out some examples, in particular the Diophantine systems of \(A^k+B^k+C^k=D^k+E^k\), \(k=1,3\).

Keywords

Simultaneous Diophantine equations Cubic Diophantine equations Equal sums of cubes 

Mathematics Subject Classification

Primary11D45 Secondary11D72 11D25 

Notes

Acknowledgements

We are very grateful to the referee for the careful reading of the paper and giving several useful comments which improved the quality of the paper.

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Copyright information

© Akadémiai Kiadó, Budapest, Hungary 2017

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of ScienceUrmia UniversityUrmiaIran
  2. 2.Department of Mathematics, Faculty of ScienceAzarbaijan Shahid Madani UniversityTabrizIran

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