3x+1 minus the
We use Conway's Fractran language to derive a function R : Z(+)-->Z(+) of the form R(n) = r(i)n if nequivalent to1 mod d where d is a positive integer, 0less than or equal toi<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-1)} of positive integers for the 3x+1 function must satisfy [GRAPHICS] where O={i: x(i) is odd}, E={i : x(i) is even}, and k=&VERBAR;O&VERBAR;. The method used illustrates a general mechanism for deriving mathematical results about the iterative dynamics of arbitrary integer functions from Fractran algorithms.
fractran collatz conjecture 3x+1 problem discrete dynamical systems