Besides the $3x + 1$ problem, for which similar problems are still unresolved regarding trayectory?

Generalize the $3x + 1$ problem as $cx \pm 1$, where $c$ is a positive odd integer and $x$ is a positive integer iterated through the function as far as possible to discover a cycle. If $x$ is even, then you halve it. But if $x$ is odd, you do either $cx + 1$ or $cx - 1$ as the case may be. (If you prefer, $c$ may be negative and you disallow $cx - 1$ for the odd branch; then $|-3x + 1|$ and $3x - 1$ are kind of the same).

With $3x - 1$ and $5x + 1$ it is somewhat well-known that many $x$ don't lead to 1, while with $3x + 1$ the question is unresolved despite intense scrutiny by many professionals and amateurs. For which other $cx \pm 1$ is the question of ultimate arrival at 1 still undetermined despite study by more than a few people? I would appreciate journal articles that look at several different $cx \pm 1$.

Solution 1:

upps:[update] I've just recently given the same answer in this question, maybe I'd delete this or the other one

• All $c$ of the form $c=2^C-1= \{3,7,15,31,63,....\}$ have a "trivial cycle" at $1 \to 1$.
• All $c$ of the form $c=2^C+1= \{3,5,9,17,33,....\}$ have a "trivial cycle" at $-1 \to -1$.
• $c=3$ has some extra cycles in the negative numbers, I've found none other than that known for instance in wikipedia
• $c=5$ has some extra cycles in the positive numbers, I've found only the two well known ones ($17 \to 43 \to 27 \to 17 $ and $13 \to 33 \to 83 \to 13 $ )
• $c=181$ ( $=\lceil 2^{15/2} \rceil-1$ ) has two cycles $27\to 611\to 27$ and $35 \to 99 \to 35$

I've searched in two modes : all odd numbers up to 9999 for $c$ checking possible cycle-lengthes up to 100 ; possible cycle lengthes up to 1000 (I think, have it not at hand at the moment) and optimal $c$ using the convergents of continued fractions excluding even extremely high $c$ requiring elements of a possible cycle $a_k$ below some upper bound taken by some formula depending on $c$ and projected cyclelength.

I've even taken negative values for $c$ and found some more cycles on small numbers but have it not at hand at the moment (but see some recent MSE-answer of mine)

Solution 2:

Here are some references to the problem $qx+1$, for example:

• R.Steiner. On the "$QX+1$ problem", $Q$ odd.. Fibonacci Quarterly, 19(3), (1981), 285-288
• R.Steiner. On the "$QX+1$ problem", $Q$ odd. II. Fibonacci Quarterly, 19(4), (1981), 293-296

Author shows that for $q=5$ there is only one non-trivial cycle ($13\to 208 \to 13$), while for $q=7$ there are no non-trivial cycles.

• R. E. Crandall. On the "$3x+1$" Problem. Mathematics of Computation, 32, (1978), 1281-1292.

Here Crandall conjectured that for all $q \geq 5$, there always exist some $n$ that never iterate to $1$.

• Z. Franco and C. Pomerance. On a Conjecture of Crandall concerning the $px+1$ Problem. Mathematics of Computation, 64(211), (1995), 1333-1336.

Here they show that Crandall conjecture is true if $q$ is a Wieferich number.

For exhaustive references on the problem I recommend to check The Ultimate Challenge: The $3x+1$ Problem by Jeffrey C. Lagarias. Point of the book is to summarize all results and references to the problem, so it seems a good place to start (the references for $qx+1$ are taken from there).