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:
\(\ds 5^2\) | \(\nmid\) | \(\ds 2465\) | ||||||||||||
\(\ds 17^2\) | \(\nmid\) | \(\ds 2465\) | ||||||||||||
\(\ds 29^2\) | \(\nmid\) | \(\ds 2465\) |
We also have that:
\(\ds 2464\) | \(=\) | \(\ds 616 \times 4\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 154 \times 16\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 88 \times 28\) |
The result follows by Korselt's Theorem.
$\blacksquare$
Sources
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $2465$