The search for self-contained numbers: k-special 3-smooth representations and the Collatz conjecture

Abstract

The Collatz conjecture is a deceptively simply problem that straddles the line between number theory and dynamical systems. It asks: if we iterate the function that sends some even n to n+2 and odd n to 3n+1, will this converge to 1 for every natural number? This problem has long stood unsolved despite attempts in many mathematical disciplines – in large part due to the difficulty of predicting the multiplicative structure of a number under addition. In this project, we provide a derivation of the most standard algebraic reformulation of the non-trivial cycles subproblem. This results in an infinite family of exponential Diophantine equations which correspond to k-special 3-smooth representations of integers. By imposing conditions on the exponents in these representations, we rewrite it in a multiplicative form that admits iterative solving for parameters of the representation. Doing so while enforcing a maximum value on the largest power of 2 in the representation, we derive a sufficient condition for no non-trivial cycles existing in this process. We show that a self-contained number, w, is exactly one which has an odd element of its orbit modularly equivalent to −3^−1 mod w. We then show that non-cyclicity of any self-contained number greater than 5 is sufficient to show that no cycles exist in the Collatz process. This differs from previous modularity-based results, and experimental results suggest that self-contained numbers are relatively rare. We show that exactly 7 such numbers exist less than 10^15 – improving on the previously known bound of 10^11.