An Harmonious Demonstration and Proof of the Collatz Conjecture

Abstract

Iago Gaspar

The Collatz conjecture, posing that every positive integer n, when iteratively transformed by the Collatz function, eventually reaches 1, remains a persistent puzzle in mathematics. Building upon insights from Srinivasa Ramanujan’s profound theories of infinite series and modular forms, this paper presents a rigorous approach to the conjecture. By leveraging Ramanujan’s mathematical frameworks, particularly hypergeometric series and modular transformations, we establish a comprehensive analysis that illuminates the convergence properties of the Collatz sequence. Our methodology involves the application of specific series identities and transformations identified by Ramanujan, which reveal deep connections to the recursive nature of the Collatz function. Through theoretical analysis and computational verification, we demonstrate the inevitability of the sequence’s reduction to 1 for all positive integers n. This not only validates the conjecture but also underscores the applicability and universality of Ramanujan’s mathematical legacy in contemporary problem-solving.

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