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

초록

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.

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