Using coding devices based on a theorem due to Zsigmondy, Birkhoff and Vandiver, we first define in terms of successor $S$ and coprimeness predicate $\perp$ a full arithmetic over the set of powers of some fixed prime, then we define in the same terms a restriction of the exponentiation. Hence we prove the main result insuring that all arithmetical relations and functions over prime powers and their opposite are $\{S, \perp\}$-definable over $\mathbf{Z}$. Applications to definability over $\mathbf{Z}$ and $\mathbf{N}$ are stated as corollaries of the main theorem.