Hello everybody,

I was playing around testing stuff with Collatz conjecture and I got the idea of a simple generalisation:

0) Pick a natural number

1) If the number is divisible by A, then divide it by A,

2) otherwise multiply it by B and add C.

3) Repeat 1-2 with the new number.

Will we always reach value 1 regardless of the initial picked value? According to Collatz conjecture we always will for A=2, B=3, C=1.

The generalisation has some constraints such as B + C ≥ 1 and A, B > 0 (so we can ensure that we stay in positive numbers) and also ignoring triples (A, B, C) such that B and C are both divisible by A, as such triple (A, B, C) is obviously gonna behave the same way as (A, B / A, C / A).

Now I wonder about triples (A, B, C) that fulfill the Collatz conjecture idea — you always end up at 1, let's call a triple that fulfills that a Collatz triple. I wanted to see if there are other Collatz triples (not prove, obviously, but experimentally check for computable values) other than the well-known (2, 3, 1).

However, as we increase the numbers they start to blow-up easily and overflow a lot, so I managed to get results for very few triples, and didn't find any new Collatz triple, so I was wondering if somebody has done a similar experiment and maybe found any?

Wikipedia briefly mentions this generalisation in 'Undecidable generalisations' claiming it's an algorithmically undecidable problem to determine whether the values always reach 1, but if I'm not mistaken this doesn't mean that other triples don't exist.

Collatz conjecture isn't one of the most useful ones but I like its simplicity so if somebody has done experiments of their own or knows some interesting published ones, I'd be glad to see the results.

Thanks in advance! :)