EMBEDDING THE 3X+1 CONJECTURE IN A 3X+D CONTEXT

Recall the well-known 3x + 1 conjecture: if T(n) = (3n + 1)/2 for n od d and T(n) = n/2 for n even, repeated application of T to any positive integer eventually leads to the cycle {1 --> 2 --> 1}. We study a nat ural generalization of the function T, where instead of 3n + 1 one tak es 3n + d, for d equal to -1 or to an odd positive integer not divisib le by 3. With this generalization new cyclic phenomena appear, side by side with the general convergent dynamics typical of the 3x + 1 case. Nonetheless, experiments suggest the following conjecture: For any od d d 2 -1 not divisible by 3 there exists a finite set of positive inte gers such that iteration of the 3x + d function eventually lands in th is set. Along with a new boundedness result, we present here an improv ed formalism, more clear-cut and better suited for future experimental research.