Generalized Solutions of the Diophantine Equation 1^n/v_1 +2^n/v_2 +3^n/v_3 +⋯+k^n/v_k =1
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Abstract
This paper investigates the positive integer solutions of the Diophantine equation where
is a positive integer and
are positive integers with the condition
. The study focuses on the existence and structural properties of the solution set. For a fixed smallest denominator, it is shown that the equation admits only finitely many solutions, and explicit upper and lower bounds for all denominators are established. Necessary conditions and polynomial growth constraints for admissible solutions are derived. These results clarify the structural behavior of power-type fractional Diophantine equations and provide a theoretical framework for further developments in number theory.
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References
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Saethao, L., Muangkarn, P. and Suanoom, C. (2024). The Solution of the Diophantine Equation 1^3/u_1+2^3/u_2+⋯+k^3/u_k=1. Journal of Science and Technology CRRU, 3(1), 1-9. https://li01.tci-thaijo.org/index.php/jstcrru/article/view/262501
