Deciphering the Collatz Conjecture Through Recursion

More than 80 years has passed since the Collatz conjecture has been proposed, but since then there has been no concrete proof as to why it is stuck in the 4 to 2 to 1 loop endlessly. There had been various attempts to find the reason as to why this infinite loop occurs, but there hasn’t been any widely recognized proof for the Collatz conjecture.

This paper details a different approach to explain why this infinite loop occurs and at the same time explain why the Collatz conjecture is similar to a computer source code. Section 2, 3, 4, and 5 details the process used to determine the purpose of 2n and n+1 functions inside an iterative loop, while section 6 and 7 reveals the logical reasoning behind how the 2n+2 algebraic expression was obtained based on 3n+1 or 2n+n+1.

Section 8 provides proof for 2n+2 as the hidden algebraic expression of the Collatz conjecture, while both sections 9 and 10 provides proof that it is also possible to get stuck in the -4 to -2 to -1 loop endlessly when n is any negative odd integer by simply replacing addition with subtraction. Sections 11 provides the key reason why the Collatz conjecture is getting stuck in an endless loop of 4 to 2 to 1.