Revealing a Singularity in Collatz Sequences

The Collatz conjecture, first proposed by Lothar Collatz in 1937, has captivated generations of mathematicians due to its deceptive simplicity and its enduring resistance to proof. Also referred to as the 3n+1 problem, the Syracuse problem, the Ulam conjecture, or the Hailstone sequence, it has spread informally across academic communities, often through oral traditionand recreational mathematics. Its basic rule can be explained to a child, yet its resolution has defied the most brilliant minds in mathematics. As Shizuo Kakutani noted in 1960, “For about a month everyone at Yale worked on it, with no result... A joke was made that this problem was part of a conspiracy to slow down mathematical research in the U.S.” Paul Erdős, in 1983, famouslydeclared that “Mathematics is not yet ready for such questions.” More recently, in 2010, Jeffrey Lagarias 1 described it as “an extraordinarily difficult problem, completely out of reach of present day mathematics. The conjecture sits at the intersection of several mathematical fields, including number theory, dynamical systems, and the study of chaotic behavior. Despite vast numerical evidence and partial results, a general proof remains elusive. In this work, we use the Hidden Order method which reveals many patterns of the Collatz sequences and, more importantly, a singularity that will radically change the understanding of the Collatz dynamic.