THE THEORY OF THE COLLATZ PROCESS T. AGAMA Abstract. In this paper we introduce and develop the theory of the Collatz process. We leverage this theory to study the Collatz conjecture. This theory also has a subtle connection with the infamous problem of the distribution of Sophie germain primes. We also provide several formulation of the Collatz conjecture in this language. 1. Introduction and motivation Recall the Collatz function, the arithmetic function of the form Definition 1.1. Let a ∈ N, then the Collatz function is the piece-wise function C (a)=  a 2 if a ≡ 0 (mod 2) 3a +1 if a ≡ 1 (mod 2). Then Collatz conjecture, which is one of the acclaimed hardest but easy to state problems is the assertion that Conjecture 1.1. Let C be the Collatz function, then min{C s (b)} ∞ s=0 = 1 for any b ∈ N. The conjecture has long been studied and hence the vast literature and surveys concerning the study. For instance the problem has been given a fair treatment in the following surveys [1], [2], [3]. Motivated by this problem we introduce the subject of the Collatz process. We develop this theory and it turns out incidentally that it is connected to other open problems such as the problem concerning the distribution of the Sophie germain primes. 2. Modified Collatz function and the Collatz process In this section we introduce s slight variant of the Collatz function and introduce the notion of the Collatz process. We introduce the notion of the backward Collatz process and the generator of the Collatz process. Definition 2.1. Let a ≥ 1, then the Collatz function is the piece-wise function f (a)=      a 2 if a ≡ 0 (mod 2),a> 1 3a +1 if a ≡ 1 (mod 2),a> 1 1 If a =1. Date : October 31, 2019. 2000 Mathematics Subject Classification. Primary 54C40, 14E20; Secondary 46E25, 20C20. Key words and phrases. Collatz; index; order; backward Collatz process. 1