About the Collatz conjecture

Abstract.

This paper refers to the Collatz conjecture. The origin and the formalization of the Collatz problem are presented in the first section, named “Introduction”. In the second section, entitled “Properties of the Collatz function”, we treat mainly the bijectivity of the Collatz function. Using the obtained results, we construct a (set of) binary tree(s) which “simulate(s)”– in a way that will be specified – the computations of the values of the Collatz function. In the third section, we give an “efficient” algorithm for computing the number of iterations (recursive calls) of the Collatz function. A comparison between our algorithm and the standard one is also presented, the first being at least 2.25 “faster” (3.00 in medium). Finally, we describe a class of natural numbers for which the conjecture is true.