Discrete Mathematics & Theoretical Computer Science
| author: | Kenneth G. Monks |
|---|---|
| title: | 3x+1 Minus the + |
| keywords: | Collatz Conjecture, 3x+1 problem, Fractran, discrete dynamical systems |
| abstract: | 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. |
| reference: | Kenneth G. Monks (2002), 3x+1 Minus the +, Discrete Mathematics and Theoretical Computer Science 5, pp. 47-54 |
| bibtex: | For a corresponding BibTeX entry, please consider our BibTeX-file. |
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| pdf-source: | dm050103.pdf (94 K) |
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