On the $\lq\lq 3x+1''$ problem

It is an open conjecture that for any positive odd integer m the function

$$C(m) = (3m + 1)/{2^{e(m)}},$$

where $e(m)$ is chosen so that $C(m)$ is again an odd integer, satisfies ${C^h}(m) = 1$ for some h. Here we show that the number of $m \leqslant x$ which satisfy the conjecture is at least ${x^c}$ for a positive constant c. A connection between the validity of the conjecture and the diophantine equation ${2^x} - {3^y} = p$ is established. It is shown that if the conjecture fails due to an occurrence $m = {C^k}(m)$, then k is greater than 17985. Finally, an analogous "$qx + r$" problem is settled for certain pairs $(q,r) \ne (3,1)$.