Walking Cautiously Into the Collatz Wilderness: Algorithmically, Number Theoretically, Randomly

Edward G. Belaga (1) , Maurice Mignotte (1)

1 IRMA - Institut de Recherche Mathématique Avancée

Building on theoretical insights and rich experimental data of our preprints, we present here new theoretical and experimental results in three interrelated approaches to the Collatz problem and its generalizations: \emphalgorithmic decidability, random behavior, and Diophantine representation of related discrete dynamical systems, and their \emphcyclic and divergent properties.

Mots clés: Diophantine approximations; exponential Diophantine equations; algorithmic decidability; pseudo-rundom walks; 3n+1; or Collatz problem; Collatz and Conway transforms; discrete dynamical systems

Domaines: Algorithme et structure de données [cs.DS]; Mathématique discrète [cs.DM]; Combinatoire [math.CO]

Edward G. Belaga, Maurice Mignotte. Walking Cautiously Into the Collatz Wilderness: Algorithmically, Number Theoretically, Randomly. Fourth Colloquium on Mathematics and Computer Science Algorithms, Trees, Combinatorics and Probabilities, 2006, Nancy, France. pp.249-260, ⟨10.46298/dmtcs.3512⟩. ⟨hal-01184717⟩