  The present work deals with the characterization of parity vectors of Collatz
sequences (of finite and infinite length). Such a characterization leads to the
determination of several numbers (integers or non-integers) that we call the
characteristic numbers of a given parity vector. Some characteristic numbers
are linked together by equations that can be called characteristic equations of
the considered parity vector. If a parity vector v of finite length n contains
the first n terms of a parity vector V of infinite length then all the
characteristic numbers of v are considered as characteristic numbers of order n
of the infinite vector V. The limits of nth order characteristic numbers when n
tends to infinity constitute the absolute characteristic numbers of the
infinite vector V and they allow determining its behavior and properties. In
this paper, we study the properties of parity vectors of infinite length based
on their characteristic numbers. Then, we establish the relations between the
first term of a Collatz sequence and its parity vector. Finally, still based on
the characteristic numbers, we determine some conditions of existence and
non-existence of divergent sequences.

General Mathematics