Does this have some significance for back propagation or something, or is it just an interesting trick of arithmetic? //not that it needs to have a technical use, it's still neat.

Collatz, busy-beavers, and algorithmic information theory are all related. To the extent they offer insight into the sparseness or density of irreducible complexity in the space of all computation.. this has many implications for what can be computed efficiently, what can be learned efficiently, program-synthesis, what can be analyzed "at a distance" without just trying it and potentially needing to wait forever, etc.

Whether it will say anything very significant practically or only philosophically is a different question. Maybe it is something like the discovery of transcendentals.. finding out that most of the number line won't have a tidy algebraic closed-form isn't exactly a make-or-break deal for the program of mathematics itself, and it also doesn't matter much to people who are doing engineering

Hailstone numbers has been a popular subject in computing circles since forever. Not much practical application, just a very simple, but curious construct.

Crazy how AI has infected every conversation these days.

> Does this have some significance for back propagation or something

You're right!

What could darn possibly matter these days, in the whole entirety of the realm Mathematics, if it does now somehow have a measurable impact on backprop ?