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

S. Subburam, Lewis Nkenyereye, N. Anbazhagan, S. Amutha, M. Kameswari, Woong Cho and Gyanendra Prasad Joshi
Additional contact information
S. Subburam: Department of Mathematics, Alagappa University, Karaikudi 630004, India
Lewis Nkenyereye: Department of Computer and Information Security, Sejong University, Seoul 05006, Korea
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

Mathematics, 2021, vol. 9, issue 15, 1-9

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 .

Keywords: Diophantine equation; Ternary Diophantine equation (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2021
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