Treffer: MONOTONIC COLLATZ SUBSEQUENCES WITH TERMS CONGRUENT MODULO A FIXED POWER OF TWO.

A class of sequences called L-sequences is introduced, each one being a subsequence of a Collatz sequence. Every ordered pair $(v,w)$ of positive integers determines an odd positive integer P such that there exists an L-sequence of length n for every positive integer n , each term of which is congruent to P modulo $2^{v+w+1}$. The smallest possible initial term of such a sequence is described. If $3^v>2^{v+w}$ the L-sequence is increasing. Otherwise, it is decreasing, except if it is the constant sequence P. A central role is played by Bezout's identity. [ABSTRACT FROM AUTHOR]