I stumbled on this collatz-like open problem recently, and was nerd-sniped into spending a lot of time thinking about. The problem is this: given the following function L, does it terminate on all inputs?
L(3k) = 0
L(3k + r) = L(5k + r + 3)
Note that it's an open problem, so finding a solution is not very likely.
(source for the problem:)
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