Carmichael Number/Examples/2465

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Example of Carmichael Number

$2465$ is a Carmichael number:

$\forall a \in \Z: a \perp 2465: a^{2465} \equiv a \pmod {2465}$

while $2465$ is composite.


Proof

We have that:

$2465 = 5 \times 17 \times 29$

and so:

\(\displaystyle 5^2\) \(\nmid\) \(\displaystyle 2465\)
\(\displaystyle 17^2\) \(\nmid\) \(\displaystyle 2465\)
\(\displaystyle 29^2\) \(\nmid\) \(\displaystyle 2465\)


We also have that:

\(\displaystyle 2464\) \(=\) \(\displaystyle 616 \times 4\)
\(\displaystyle \) \(=\) \(\displaystyle 154 \times 16\)
\(\displaystyle \) \(=\) \(\displaystyle 88 \times 28\)

The result follows by Korselt's Theorem.

$\blacksquare$


Sources