The following was closed and deleted, presumably because is contains two exercises, and a cursory reading might place it, therefore, as needing focus; it does not: it's about an apparent contradiction in Robinson's book on group theory between the two exercises. https://math.stackexchange.com/q/4...

Just wondering if this would be both necessary and sufficient for a proof. First, define a Collatz representation of an odd number $n$ in the form: $$2^{\sum_{i=1}^k{a_i}}-\sum_{i=0}^{k-1}3^i2^{\sum_{j=i+2}^k a_{j}} - 3^{k-1} = 3^kn$$ Here, $k$ is the number of steps in the reduced form algorithm...