Fermat's Little Theorem/Corollary 3

Corollary to Fermat's Little Theorem

Let $p^k$ be a prime power for some prime number $p$ and $k \in \Z_{\gt 0}$.

Then:

$\forall n \in \Z_{\gt 0}: n^{p^k} \equiv n \pmod p$

Proof

The proof proceeds by induction.

For all $k \in \Z_{\ge 1}$, let $P \paren {k}$ be the proposition:

$\forall n \in \Z_{\gt 0}: n^{p^k} \equiv n \pmod p$

Basis for the Induction

$P \paren {1}$ is the case:

$\forall n \in \Z_{\gt 0}: n^p \equiv n \pmod p$

which follows from the corollary 1 to Fermat's Little Theorem.

This is the basis for the induction.

Induction Hypothesis

Now it needs to be shown that, if $P \paren{k-1}$ is true, where $k \ge 2$, then it logically follows that $P \paren {k}$ is true.

So this is the induction hypothesis:

$\forall n \in \Z_{\gt 0}: n^{p^{k - 1} } \equiv n \pmod p$

from which it is to be shown that:

$\forall n \in \Z_{\gt 0}: n^{p^k} \equiv n \pmod p$

Induction Step

This is the induction step:

For any $n \in \Z_{\gt 0}$ then:

\(\displaystyle n^{p^k}\)

\(=\)

\(\displaystyle \paren {n^{p^{k - 1} } }^p\)

$\quad$

$\quad$

\(\displaystyle \)

\(\equiv\)

\(\displaystyle n^{p^{k - 1} } \pmod p\)

$\quad$ Corollary 1 to Fermat's Little Theorem

$\quad$

\(\displaystyle \)

\(\equiv\)

\(\displaystyle n \pmod p\)

$\quad$ Induction Hypothesis

$\quad$

$\blacksquare$