PRH | Aux | 4.3.3 • Refreshers for the Collatz Certificate Method

These exercises refresh the exact techniques used in the Lyapunov-certificate proof for the accelerated Collatz map: 2 -adic valuation and odd part, residue arithmetic modulo $2^k$, the exceptional residue, the accelerated step and its $\log$ decomposition, basic bounds for $\ln$ and telescoping, and the max-plus/difference-constraints viewpoint (cycle means, Bellman-Ford). Solutions are brief and to the point. Throughout, $k \geq 1, M=2^k, S=\left\{1,3, \ldots, 2^k-1\right\}$, and for odd $N$ we set $$F^{\sharp}(N)=\frac{3 N+1}{2^{v_2(3 N+1)}}=\operatorname{odd}(3 N+1), \quad \varepsilon(N)=\ln \left(1+\frac{1}{3 N}\right) .$$