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On the Ternary Exponential Diophantine Equation Equating a Perfect Power and Sum of Products of Consecutive Integers

Author

Listed:
  • S. Subburam

    (Department of Mathematics, Alagappa University, Karaikudi 630004, India
    These authors contributed equally to this work.)

  • Lewis Nkenyereye

    (Department of Computer and Information Security, Sejong University, Seoul 05006, Korea
    These authors contributed equally to this work.)

  • N. Anbazhagan

    (Department of Mathematics, Alagappa University, Karaikudi 630004, India)

  • S. Amutha

    (Ramanujan Centre for Higher Mathematics, Alagappa University, Karaikudi 630003, India)

  • M. Kameswari

    (Department of Mathematics, School of Advanced Sciences, Kalasalingam Academy of Research and Education, Krishnankoil, Srivilliputhur 626128, India)

  • Woong Cho

    (Department of Automotive ICT Convergence Engineering, Daegu Catholic University, Gyeongsan 38430, Korea)

  • Gyanendra Prasad Joshi

    (Department of Computer Science and Engineering, Sejong University, Seoul 05006, Korea)

Abstract

Consider the Diophantine equation y n = x + x ( x + 1 ) + ⋯ + x ( x + 1 ) ⋯ ( x + k ) , where x , y , n , and k are integers. In 2016, a research article, entitled – ’power values of sums of products of consecutive integers’, primarily proved the inequality n = 19,736 to obtain all solutions ( x , y , n ) of the equation for the fixed positive integers k ≤ 10 . In this paper, we improve the bound as n ≤ 10,000 for the same case k ≤ 10 , and for any fixed general positive integer k , we give an upper bound depending only on k for n .

Suggested Citation

  • S. Subburam & Lewis Nkenyereye & N. Anbazhagan & S. Amutha & M. Kameswari & Woong Cho & Gyanendra Prasad Joshi, 2021. "On the Ternary Exponential Diophantine Equation Equating a Perfect Power and Sum of Products of Consecutive Integers," Mathematics, MDPI, vol. 9(15), pages 1-9, July.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:15:p:1813-:d:605564
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