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Then ϕ m+1 (1) = = ϕ(1 + (h + th + t 2 h + · · · + t m−1 h)) = ϕ((h + th + t 2 h + · · · + t m−1 h) + 1) = = ϕ(h + th + t 2 h + · · · + t m−1 h) + ϕ(1) = th + t 2 h + · · · + t m h + 1 + h = = 1 + h + th + t 2 h + · · · + t m h, which proves the induction claim. Plugging the parameters (n; b, t, h, p) into the equations (8) and (7) allows for a deterministic calculation of ϕ(a), for all a ∈ Zn . Thus, ϕ is uniquely determined by the parameters (n; b, t, h, p). Finally, if the five-tuples associated with two skew-morphisms ϕ, ϕ differ in at least one number, it is easy to see that the two skew-morphisms differ as permutations.
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