3x+1 Minus the + - Inria - Institut national de recherche en sciences et technologies du numérique Access content directly
Journal Articles Discrete Mathematics and Theoretical Computer Science Year : 2002

3x+1 Minus the +


We use Conway's \emphFractran language to derive a function R:\textbfZ^+ → \textbfZ^+ of the form R(n) = r_in if n ≡ i \bmod 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-1 \ of positive integers for the 3x+1 function must satisfy \par ∑ _i∈ \textbfE \lfloor x_i/2 \rfloor = ∑ _i∈ \textbfO \lfloor x_i/2 \rfloor +k. \par where \textbfO=\ i : x_i is odd \ , \textbfE=\ i : x_i is even \ , and k=|\textbfO|. \par The method used illustrates a general mechanism for deriving mathematical results about the iterative dynamics of arbitrary integer functions from \emphFractran algorithms.
Fichier principal
Vignette du fichier
dm050103.pdf (56.26 Ko) Télécharger le fichier
Origin : Files produced by the author(s)

Dates and versions

hal-00958971 , version 1 (13-03-2014)



Kenneth G. Monks. 3x+1 Minus the +. Discrete Mathematics and Theoretical Computer Science, 2002, Vol. 5, pp.47-54. ⟨10.46298/dmtcs.297⟩. ⟨hal-00958971⟩


69 View
775 Download



Gmail Facebook X LinkedIn More