Fermat's Little Theorem/Corollary 2

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Corollary to Fermat's Little Theorem

Let $p$ be a prime number.

Then:

$n^{p - 1} \equiv \sqbrk {p \nmid n} \pmod p$

where:

$\nmid$ denotes non-divisibility
$\sqbrk \cdots$ is Iverson's convention.


Proof

If $p \nmid n$ then from Fermat's Little Theorem:

$n^{p - 1} \equiv 1 \pmod p$

If $p \divides n$ then:

$p \divides n^{p - 1}$

and $n^{p - 1} \equiv 0 \pmod p$ by definition.

Hence the result by definition of Iverson's convention.

$\blacksquare$


Source of Name

This entry was named for Pierre de Fermat.


Sources