Power Function is Completely Multiplicative/Integers
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Theorem
Let $c \in \Z$ be an integer.
Let $f_c: \Z \to \Z$ be the mapping defined as:
- $\forall n \in \Z: \map {f_c} n = n^c$
Then $f_c$ is completely multiplicative.
Proof
Let $r, s \in \Z$.
Then:
\(\ds \map {f_c} {r s}\) | \(=\) | \(\ds \paren {r s}^c\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds r^c s^c\) | Product of Powers | |||||||||||
\(\ds \) | \(=\) | \(\ds \map {f_c} r \map {f_c} s\) |
$\blacksquare$
Sources
- 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.4$: Integer Functions and Elementary Number Theory: Exercise $29 \ \text{(a)}$