Collatz Conjecture: Properties of odd integers that make up a cycle

Solution 1:

You make an induction on the length of a cycle. And then in the induction step you even assume that the same numbers $x_i$ of a $k$-cycle are also in a $(k+1)$-cycle. Thus you essentially assume that the cycle has length $1$.

Edit after revision of the question: The claim $$ 2^{L_k}x_{i+k} = 3^kx_i + 3^{k-1} + \sum_{s=1}^{k-1}3^{k-1-s}2^{\sum_{t=0}^s\nu_2(3x_{i+t}+1)}$$ is verifiably wrong, e.g., for $k=2$, $x_1=27$, $x_2=\frac{3\cdot 27+1}2=41$, the right hand side evaluates to $$9\cdot 27+3+2^{\nu_2(3\cdot 27+1)+\nu_2(3\cdot 41+1)} =254,$$ which is not even a multiple of $x_3=\frac{3\cdot 41+1}{4}=31$.