TITLE: 3x+1 minus the +

AUTHOR: Ken Monks
        Department of Mathematics
        University of Scranton
        Scranton, PA 18510
        email: monks@scranton.edu

ABSTRACT:

We use Conway's Fractran language to derive a function R:Z->Z of the
form

         R(n)=r_i*n if n=i mod d

where d is a positive integer, 0<=i<d and r_0,r_1,...,r_{d-1} are rational 
numbers, such that the famous 3x+1 conjecture holds if and only if the R-orbit 
of 2^n contains 2 for all positive integers n. We then show that the R-orbit 
of an arbitrary positive integer is a constant multiple of an orbit that 
contains a power of 2. Finally we apply our main result to show that any 
cycle {x_0,...,x_m} of positive integers for the 3x+1 function must satisfy

  sum_{i in E} floor(x_i/2)=sum_{i in O} floor(x_i/2) + k.

where O={ i : x_i is odd}, E={ i : x_i is even}, and
k=|O|. The method used illustrates a general mechanism for deriving 
mathematical results about the iterative dynamics of arbitrary 
integer functions from Fractran algorithms.