Sur la conjecture de Collatz

TL;DR

This paper generalizes the Collatz conjecture to the 2-adic numbers, exploring average behavior of sequences and proposing a broader framework involving pairs (p,q) under certain conditions.

Contribution

It extends the Collatz conjecture to 2-adic completions and introduces a new approach using isometries of $Q_2$ to analyze sequence behavior.

Findings

Provides a generalized Collatz conjecture on 2-adic numbers.

Shows how isometries of $Q_2$ inform sequence behavior.

Suggests possible generalizations to pairs (p,q) under specific inequalities.

Abstract

We give a generalization of Collatz conjecture or 3n+1 problem on 2-adic completion of Q. A isometric of $Q_2$ provides information on the average behavior of the firsts terms of the sequence according to the class of $u_0$ modulo $2^m$. A generalisation from the pair (2,3) to the pair (p,q) seems possible under the assumption $q^{p-1}<p^p$.