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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 $n$?
joriki
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DaBler
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    "no infinite cyclic trajectories" huh? A trajectory might be infinite, or it might be cyclic, but how can it be both? Anyway, so far as I know, there is no solved example of any generalized Collatz question. The go to source is the book by Lagarias. – Gerry Myerson Jul 03 '18 at 12:41
  • The version $7n+1$ for odd $n$ and $n/2$ for even $n$ seems to diverge already for start value $3$ – Peter Jul 03 '18 at 13:17
  • @Peter Yes, you are right. This is actually the case discussed in the paper of Crandall above. One can, however, consider more general form $f(n) = 7n+r$ if $n \equiv 1 \pmod{2}$. My question is whether any variant of this problem had been addressed in the literature. – DaBler Jul 03 '18 at 16:36
  • @GerryMyerson The paper discusses two types of trajectories – bounded and unbounded (diverging). The bounded can contain cycles (thus infinite cyclic trajectory). Anyway, I am aware of the existence of the book by Lagarias. Unfortunately, I do not have it available at the moment. – DaBler Jul 03 '18 at 19:52
  • What is easy to show is that it has the trivial cycle $1 \to 1 \to \cdots$ , I've not found any other cycles (even tried in the negative numbers). So it seems, that all numbers which do not decline to $1$ diverge... (Note that this is similar to all forms $mx+1$ where $m$ is of the form of $m=2^A-1={3,7,15,31,63, \ldots }$ ) – Gottfried Helms Jul 03 '18 at 20:19
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    Just saw this: https://arxiv.org/abs/1807.00908 – gammatester Jul 04 '18 at 07:42
  • For the sake of completeness, I also refer to two similar questions (not duplicates): https://math.stackexchange.com/q/1408656 and https://math.stackexchange.com/q/2808138 – DaBler Jul 04 '18 at 22:11
  • @GottfriedHelms Could you consider the function $f(n) = 7n \pm 1$ as defined in https://arxiv.org/abs/1807.00908. The trajectory for any initial positive integer eventually reaches the cycle 1, 8, 4, 2, 1... I would appreciate hearing any thoughts on this. – DaBler Jul 06 '18 at 08:52
  • @DaBler: very nice! (Just tried some small values; of some guessed interest are perhaps initial values being odd multiples of $7$ because they have no odd precedessor, $5*7=35$ has a "long" trajectory) I'll see what I can say about it using my analytical toolbox later or tomorrow... – Gottfried Helms Jul 06 '18 at 12:31
  • @DaBler - reading the arxiv-article now in full I'm really frustrated about the tendency of the author to insist on writing about a $7x+1$ problem contrary to of what he is really looking at: a $7x \pm 1$ - problem. I think at least one time he correctly took that latter fomulation. I've really no idea why to force the impression one had results on the $7x+1$ problem other than the known ones, and why to introduce such a fuzzyness (in the already often fuzzily discussed Collatz-problem ... ). For me it is a shadow on the really great basic observation! – Gottfried Helms Jul 07 '18 at 06:54
  • @GottfriedHelms In such a case, the name should be the $7n\pm1$ problem, rather than $7x\pm1$. Shouldn't it? Anyway, I was hoping for some comments on the function itself, not its name :) – DaBler Jul 08 '18 at 10:23
  • @GerryMyerson by an infinitely cyclic trajectory the OP means a cyclic trajectory rather than an infinitely non-repeating one. In the sense that every cyclic trajectory is infinite - that's how it can be both. – Robert Frost Jul 09 '18 at 14:05
  • There is a link to an interactive graph showing the orbits of all numbers under the 7x±1 mapping: https://wavelets.org/collatz/collatz-graph/ – DaBler Jul 27 '18 at 20:30

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