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There are all kinds of execution variants to the collatz conjecture for when hitting an odd number: $3n+1$ or $3n+3^a$ or $1.5n + 0.5$ or $1.5n + 1.5$... . The assumption is: proving any of them will prove all of them.

I have been experimenting with different executions and stumbled upon the following: if divisible by $3$: divide by $3$, else if divisible by $2$: divide by $2$, else: execute $5n +1$. see results here

It seems to follow the same usual collatz patterns of hitting $1$ eventually, whereas any other variant such as $7n+1$, & $11n +1$... seem to have infinite sequences (never hitting $1$).

My question is:

If the original conjecture ($3n+1$) is proven, will it also prove what I have stumbled upon or will it be a totally different conjecture?

Update: I am well aware of the Collatz patterns for when there are only 2 rules, however in my experiment there are 3 rules and if the third rule is: execute $5n +1$, all sequences seem to collapse all the way to $1$.

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  • $\begingroup$ Hard to say without seeing the proof. The basic questions are which are the key things that make the proof work? Do those things apply to your modified Collatz sequence? As you can imagine, this is not something that can be answered without having a proof of Collatz in front of you. $\endgroup$
    – Anthony Quas
    Commented Sep 8, 2020 at 4:08
  • $\begingroup$ I have attached a link for the first 200 integers, however it seems consistent for the first 1000 integers (I can't submit it because the script keeps braking) $\endgroup$
    – EMN
    Commented Sep 8, 2020 at 4:13
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    $\begingroup$ For any piecewise linear Collatzish problem, there's a simple heuristic which suggests whether sequence goes off to infinity or collapses down to small cycles. There is plenty of literature to consult on this. $\endgroup$
    – Gerry Myerson
    Commented Sep 8, 2020 at 10:51
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    $\begingroup$ It's very difficult to say unless two problems are provably equivalent. For instance it was widely expected that the twin prime problem and the binary Goldbach problem are basically equivalent (influenced by J.R. Chen's monumental work which used the same method to obtain roughly the same almost-result for both), but perhaps this has changed since we now know bounded gaps between primes while similar progress towards Goldbach has not been made. $\endgroup$
    – Stanley Yao Xiao
    Commented Sep 8, 2020 at 11:34
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    $\begingroup$ Collatz-like problems with three or four or any finite number of rules, have been studied, heuristics formulated, etc. The literature is enormous, please have a look at it. You can start with Jeff Lagarias' book, it will point you toward other resources. Also, if you want to be sure I see a comment intended for me, you have to put @Gerry in it. $\endgroup$
    – Gerry Myerson
    Commented Sep 9, 2020 at 2:38

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