## Discrete Mathematics & Theoretical Computer Science, Vol 5 (2002)

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DMTCS vol 5 no 1 (2002), pp. 47-54

# Discrete Mathematics & Theoretical Computer Science

## Volume 5 n° 1 (2002), pp. 47-54

author: Kenneth G. Monks 3x+1 Minus the + Collatz Conjecture, 3x+1 problem, Fractran, discrete dynamical systems We use Conway's Fractran language to derive a function R:Z+ → Z+ of the form R(n) = rin  if  n ≡ i &mod; d where d is a positive integer, 0 ≤ i < d and r0,r1, ... rd-1 are rational numbers, such that the famous 3x+1 conjecture holds if and only if the R-orbit of 2n 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 { x0, ... ,xm-1 } of positive integers for the 3x+1 function must satisfy ∑i∈ E ⌊ xi/2 ⌋ = ∑i∈ O ⌊ xi/2 ⌋ +k. where O={ i : xi  is odd } , E={ i : xi  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. If your browser does not display the abstract correctly (because of the different mathematical symbols) you can look it up in the PostScript or PDF files. Kenneth G. Monks (2002), 3x+1 Minus the +, Discrete Mathematics and Theoretical Computer Science 5, pp. 47-54 For a corresponding BibTeX entry, please consider our BibTeX-file. dm050103.ps.gz (0 K) dm050103.ps (93 K) dm050103.pdf (94 K)