The Collatz conjecture which is defined by: starting from all positive integer n, we apply these 2 instructions, if it is even divide it by 2, and if it is odd we multiply it by 3, we add 1 and we divide all by 2, the conjecture states that if we repeat the transformation of n, we always end up falling on 1 which in turn finds itself trapped indefinitely in the trivial cycle 1, 2, 3, 4. three steps were taken to arrive at its demonstrations, the first led to the uniqueness of the Trivial Cycle which consisted, in PART 1, of exploring in the Trivial Cycle a property recurring in all other cycles with more than three terms of which they can only generate, according to their Cycle equation, positive integers 1, 2 or 4. The second step consists of constructing a direct equivalence between the convergence of Syracuse towards 1 with the fact that the ratio of odd terms to the number of even terms is stuck in the interval [0, 0.63092975..[, and that thanks to this proposition, we were able to demonstrate the decrease of all n towards 1, finally the third step, which by a property of the average demonstrated by recurrence the equation which connects the powers of 2 with the powers of odd terms to which we applied the —Collatz 3n +1 instruction.