Academic Journal of Mathematical Sciences, 2025, 6(1); doi: 10.25236/AJMS.2025.060104.
Chao Chen
Fengxiang District Township Water Supply Management Station, Baoji, Shaanxi, 721400, China
The Collatz Conjecture primarily explores whether a sequence of positive integers, generated by a specific iterative rule, ultimately converges to 1. The study of its proof is not only aimed at answering a specific mathematical question but also at delving into the essence and boundaries of mathematics, thereby advancing the development of mathematical science. The conjecture encompasses all positive integers, focusing on understanding and proving that when performing the following operations on any positive integer: if it is odd, multiply by 3 and add 1; if it is even, divide by 2, after a finite number of iterations, each sequence of operations inevitably reaches the number 1 and forms a trivial cycle: {4, 2, 1}. By introducing the concept of roots and utilizing constructive methods and mathematical induction, we explore and analyze related issues of the Collatz Conjecture, leading to an in-depth investigation that proves the conclusion that the Collatz Conjecture transforms any positive integer into 1.
Collatz Conjecture; branch number; transformation symbols; transformation paths; roots
Chao Chen. Proof of the Collatz Conjecture. Academic Journal of Mathematical Sciences (2025) Vol. 6, Issue 1: 27-44. https://doi.org/10.25236/AJMS.2025.060104.
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