Carmichael Number/Examples/1105
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Example of Carmichael Number
$1105$ is a Carmichael number:
- $\forall a \in \Z: a \perp 1105: a^{1105} \equiv a \pmod {1105}$
while $1105$ is composite.
Proof
We have that:
- $1105 = 5 \times 13 \times 17$
and so:
\(\ds 5^2\) | \(\nmid\) | \(\ds 1105\) | ||||||||||||
\(\ds 13^2\) | \(\nmid\) | \(\ds 1105\) | ||||||||||||
\(\ds 17^2\) | \(\nmid\) | \(\ds 1105\) |
We also have that:
\(\ds 1104\) | \(=\) | \(\ds 276 \times 4\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 92 \times 12\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 69 \times 16\) |
The result follows by Korselt's Theorem.
$\blacksquare$