Embedding the 3x + 1 Conjecture in a 3x + d Context

Abstract

Recall the well-known 3x + 1 conjecture: if T(n) = (3n + 1)/2 for n odd 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 natural generalization of the function T, where instead of 3n + 1 one takes 3n + d, for d equal to −1 or to an odd positive integer not divisible 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 odd d ≥ −1 not divisible by 3 there exists a finite set of positive integers such that iteration of the 3x + d function eventually lands in this set.

Along with a new boundedness result, we present here an improved formalism, more clear-cut and better suited for future experimental research.