Fermat's Little Theorem/Corollary 3

From ProofWiki
Jump to navigation Jump to search

Corollary to Fermat's Little Theorem

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


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


The proof proceeds by induction.

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

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

Basis for the Induction

$\map P 1$ 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 $\map P {k - 1}$ is true, where $k \ge 2$, then it logically follows that $\map P k$ 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:

\(\ds n^{p^k}\) \(=\) \(\ds \paren {n^{p^{k - 1} } }^p\)
\(\ds \) \(\equiv\) \(\ds n^{p^{k - 1} } \pmod p\) Corollary 1 to Fermat's Little Theorem
\(\ds \) \(\equiv\) \(\ds n \pmod p\) Induction Hypothesis