Power Function is Completely Multiplicative/Integers

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 $29 \ \text{(a)}$