Since its proposal in 1937 by Lothar Collatz, the Collatz conjecture has become a nightmare for mathematicians. Its extremely simple statement, combined with its extraordinary difficulty, challenges even the most brilliant minds. In 1983, Paul Erdős famously declared that “Mathematics is not yet ready for such questions.” More recently, in 2010, Jeffrey Lagarias described it as “an extraordinarily difficult problem, completely out of reach of present-day mathematics.”, the latest major progress dates back to 2019, with Terence Tao’s work Almost all orbits of the Collatz map attain almost bounded values 1. One of the reasons why this problem is believed to be beyond our current capabilities is that the Collatz sequence is “too local”; that is, we lacka global algebraic structure that would force all integers to descend toward 1. In this work, we present the arguments that lead us to adopt the singularity as a binary structural invariant. We show that reaching singularity is the only gateway for escaping the growth–decay cycle of the Collatz sequence, and that convergence to 1 is nothing more than a trivial corollary.
Ammar HAMDOUS (Wed,) studied this question.