Visualizing and Proving the Collatz Conjecture: Ananta-Graph Approach

The Collatz Conjecture, proposed by Lothar Collatz in 1937, is an unsolved mathematical problem often called the “3x+1 problem” or the “Hailstone sequence”. It suggests that starting with any positive integer and applying a specific process: dividing an even number by 2 or multiplying an odd number by 3 and adding 1 will eventually result in the sequence reaching the cycle 4, 2, 1 regardless of the starting value. Despite its simplicity, no one has been able to prove or disprove this conjecture for all positive numbers. The behaviour of the Collatz Conjecture can also be illustrated through a graphical representation, showing how the sequence for different starting numbers evolves before reaching the repeating cycle of 4, 2, 1. In this paper, we highlight the various graphical representations of the Collatz sequence and finally provide the proof of Collatz conjecture by plotting the Ananta-graphs.