The Collatz conjecture is one of the simplest yet most mysterious problems in mathematics. It begins with any positive integer and follows a very basic rule: if the number is even, divide it by two; if it is odd, multiply it by three and add one. Then repeat the process with the resulting number.
Despite the simplicity of this procedure, the conjecture makes a bold claim: no matter which positive integer you start with, the sequence will always eventually reach 1. For example, starting with 6 produces the sequence 6 → 3 → 10 → 5 → 16 → 8 → 4 → 2 → 1.
What makes the Collatz conjecture fascinating is the gap between its elementary definition and its extreme difficulty. It has been tested for enormous ranges of numbers using computers, yet no general proof—or counterexample—has ever been found. As a result, it remains an open problem and a striking example of how deep complexity can arise from simple rules.
I have developed a preliminary proof sketch for the conjecture, building on foundational insights by Terence Tao and other leading mathematicians in the field. It is very likely that this sketch contains gaps or errors, as I approach the problem as an enthusiastic amateur rather than with the depth of expertise of established researchers.
The paper can be found here:
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