A noniterative $2$-adic statement of the $3N+1$ conjecture

Abstract:

Associated with the $3N + 1$ problem is a permutation $\Phi$ of the 2-adic integers. The $3N + 1$ conjecture is equivalent to the conjecture that 3Q is an integer if $\Phi (Q)$ is a positive integer. We state a new definition of $\Phi$. To wit: Q and $N = \Phi (Q)$ are linked by the equations $Q = {2^{{d_0}}} + {2^{{d_1}}} + \cdots$ and $N = ( - 1/3){2^{{d_0}}} + ( - 1/9){2^{{d_1}}} + ( - 1/27){2^{{d_2}}} + \cdots$ with $0 \leq {d_0} < {d_1} < \cdots$. We list four applications of this definition.