2 research outputs found
Clusters of Integers with Equal Total Stopping Times in the 3x + 1 Problem
The clustering of integers with equal total stopping times has long been
observed in the 3x + 1 Problem, and a number of elementary results about it
have been used repeatedly in the literature. In this paper we introduce a
simple recursively defined function C(n), and we use it to give a necessary and
sufficient condition for pairs of consecutive even and odd integers to have
trajectories which coincide after a specific pair-dependent number of steps.
Then we derive a number of standard total stopping time equalities, including
the ones in Garner (1985), as well as several novel results
The 3x+1 problem: An annotated bibliography (1963--1999) (sorted by author)
The 3x+ 1 problem concerns iteration of the map on the integers given by T(n)
= (3n+1)/2 if n is odd; T(n) = n/2 if n is even. The 3x+1 Conjecture asserts
that for every positive integer n > 1 the forward orbit of n under iteration by
T includes the integer 1. This paper is an annotated bibliography of work done
on the 3x+1 problem and related problems from 1963 through 1999. At present the
3x+1 Conjecture remains unsolved.Comment: 74 pages latex; 197 references, second title change to distinguish
from 3x+1 book; first title change indicates abridgment of earlier versions
to papers 1999 and earlier ; part II now covers papers 2000 and later, see
arxiv:math.NT/0608208; v.11 cutoff date changed from 2000 to 1999, v.13 added
Oulipo reference