THE CONGRUENCE STRUCTURE OF THE 3x + 1 MAP

Let T : Z→ Z be defined by T (x) = 12x if x is even, and T (x) = 1 2 (3x + 1) if x is odd. The 3x + 1 Conjecture asserts that every positive x has an iterate T(x) = 1. It is known that T maps congruence classes with modulus 2 to those with modulus equal to a power of 3. We describe properties of the image class residues and use those properties to show that, under iteration by T , any congruence class with modulus of the form 23 generates all integers not divisible by 3. This has negative implications for the study of sufficient sets for the 3x + 1 Conjecture. The analysis also provides insight into a particular permutation function associated with T .