Solving the Collatz Conjecture, Using Gaussian Arithmetic

We prove that the Collatz, 3n+1 , conjecture is true. To prove this statement, we will first see that, if a number n starts a sequence that satisfies the Collatz conjecture then all sequences containing that number n satisfy the conjecture. We will call these sequences Collatz trajectories. Then, we will study trajectories over equivalence classes defined by the congruences modulo 6, which were already discussed by Carl F. Gauss in 1798, and we will show that all Collatz trajectories contain some number belonging to the class [ 4 ] 6 . Finally, we will prove that all numbers belonging to class [ 4 ] 6 initiate trajectories that satisfy the conjecture. All this proves the first statement: The Collatz conjecture is true.