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Abstract
Consider the Diophantine equation yn=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.
Highlights
Received: 30 June 2021Accepted: 27 July 2021Published: 30 July 2021Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.In 1976, Tijdeman proved that all integral solutions ( x, y, n), n > 0 and |y| > 1, of the equation yn = f ( x )satisfy n < c0, where c0 is an effectively computable constant depending only on f if f ( x )is an integer polynomial with at least two distinct roots (Shorey-Tijdeman [1], Tijdeman [2], Waldschmidt [3])
This article implied a method to obtain an upper bound for all n where ( x, y, n) is an integral solution of (1) and to improve the method and algorithm of [4]
I) 657876570 vide letter No SR/FIST/MS-I/2018/17 Dt. 20 December 2018
Summary
Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. In. Theorem 2.1 of [5], Hajdu, Laishram, and Tengely proved that there exists an effectively computable constant c(k ) depending only on k such that ( x, y, n) satisfy n ≤ c(k) if y 6= 0, −1. The result of Hajdu, Laishram, and Tengely in [5] is much stronger than the following corollary. They explicitly obtained all solutions for the values k ≤ 10 using the MAGMA computer program along with two well-known methods (See Subburam [6], Srikanth and Subburam [13], and Subburam and Togbe [14]), after proving that n ≤ 19,736 for 1 ≤ k ≤ 10. All integral solutions ( x, y, 2) of (1) satisfy x ∈ H1 or min H2 ≤ x ≤ max H2
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