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 $$\{1\to2\to1\}\hbox{.}$$ 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 $d \ge -1$ 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.