Integer patterns in Collatz sequences

Abstract

The Collatz conjecture asserts that repeatedly iterating $f(x)=(3x+1)/2^{a(x)}$, where $a(x)$ is the highest exponent for which $2^{a(x)}$ exactly divides $3x+1$, always lead to $1$ for any odd positive integer $x$. Here, we present an arborescence graph constructed from iterations of $g(x)=(2^{e(x)}x-1)/3$, which is the inverse of $f(x)$ and where $x\not\equiv [0{]}_3$ and $e(x)$ is any positive integer satisfying $2^{e(x)}x-1\equiv [0{]}_3$, with $[0{]}_3$ denoting $0(\mathrm{mod}3)$. The integer patterns inferred from the resulting arborescence provide new insights into proving the validity of the conjecture.

13 pages, 2 figures; corrected typos