For the $3n-1$-problem, where the only known cycle is $\{1,2\}$, it is known that any other cycle must start after $2^{40}$, and have length $17 087 915b + kc$, where $b,c\in\mathbb{N}$ and $k\in\{301 994 , 85 137 581\}$: see Shalom Eliahou, The 3x + 1 problem: new lower bounds on nontrivial cycle lengths.Discrete Math. 118 (1993), 45–56.