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