In computational complexity Collatz and related conjectures are often used when studying computation complexity. The connection is made through the Busy Beaver function, where BB(n) is the maximum number of steps taken by any n state Turing machine that halts. There is a 15 state Turing machine that halts if and only if a conjecture by Paul Erdős (closely related to the Collatz conjecture) is false. Hence if BB(15) was known, and this machine did not stop in that number of steps, it would be known to run forever and hence no counterexamples exist (which proves the conjecture true). This is a completely impractical way to settle the conjecture; instead it is used to suggest that BB(15) will be very hard to compute, at least as difficult as settling this Collatz-like conjecture.
in computational complexity collatz and related conjectures are often used when studying computation complexity the connection is made through the busy beaver function where bb(n) is the maximum number of steps taken by any n state turing machine that halts there is a 15 state turing machine that halts if and only if a conjecture by paul erdős (closely related to the collatz conjecture) is false hence if bb(15) was known and this machine did not stop in that number of steps it would be known to run forever and hence no counterexamples exist (which proves the conjecture true) this is a completely impractical way to settle the conjecture; instead it is used to suggest that bb(15) will be very hard to compute at least as difficult as settling this collatz-like conjecture