All solutions of consecutive natural numbers sum equation and their closed forms

Purpose: This study aims to find a closed-form solution for all ordered pairs of natural numbers (?,?) satisfying the consecutive natural number sum equation 1 + 2 + ⋯ + ? sama dengan (? + 1) + (? + 2) + ⋯ + ?. This research contributes to number theory, particularly in the context of Diophantine equations.Method: The problem is solved analytically using Pythagorean triple theory and Pell's Theorem, which provides a rigorous mathematical framework for deriving solutions.Findings: The study reveals that there are infinitely many ordered pairs (?,?) of natural numbers that satisfy the equation. Furthermore, the solutions can be expressed in closed-form expressions for (??,??), where ? anggota ?, as follows:and Additionally, the ratio ??/?? approaches 1/akar2 as ? tends to infinity.Significance: The closed-form solutions provided in this study not only enhance our understanding of the consecutive natural number sum equation but also open avenues for further research in number theory fields, especially involving Diophantine equations. The findings have potential applications in theoretical and applied mathematics.