The first time I faced Syracuse conjecture I thought it was easier to start from any number n and to arrive to ni < n rather than fall down to 1. In this way I could not take into consideration even numbers, because if n is even then n ? n/2 < n. So I had only to examine the odd numbers. In addition to the proof proposed by me, I have discovered many properties and peculiarities of this famous conjecture. It hides the magical harmony of odd numbers, and may be a type of law on the expansion of Cosmos based on the power of 2, as prophesied by Plato in some of his writings. So my work takes on a popular and didactic value of this marvelous conjecture. In this paper I have only used arithmetic and elementary number theory, but, in spite of its simple enunciation, Syracuse Conjecture is a difficult topic, therefore this article needs a lot of patience in reading for a well-understanding. I have considered that various applications and examples were needed for better explain my work.
