One of the most famous problems in mathematics that remains unsolved is the Collatz conjecture.
I am concerned with similar 7x+1 problem.
I have already seen this problem mentioned in the literature. Particularly, Crandall (1978) pp. 1291–1292 states that
The outstanding unsolved case is the "7x + 1" problem, for which there may be no infinite cyclic trajectories.
So my question is:
Is this still an unresolved problem?
Is there any other literature dealing with this problem?
Possibly another question:
- Are there any other functions of the form (or similar) $$ f(n) = \begin{cases} mn + r & \text{ if $n \equiv 1 \pmod{2}$,} \\ n/2 & \text{ if $n \equiv 0 \pmod{2}$,} \\ \end{cases} $$ which return to 1 starting from any positive initial value $n$?