Carmichael Number/Examples/41,041
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Example of Carmichael Number
$41 \, 041$ is a Carmichael number:
- $\forall a \in \Z: a \perp 41 \, 041: a^{41 \, 041} \equiv a \pmod {41 \, 041}$
while $41 \, 041$ is composite.
Proof
We have that:
- $41,041 = 7 \times 11 \times 13 \times 41$
and so:
\(\ds 7^2 \times 827 + 28\) | \(=\) | \(\ds 41 \, 041\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds 7^2\) | \(\nmid\) | \(\ds 41 \, 041\) | |||||||||||
\(\ds 11^2 \times 339 + 22\) | \(=\) | \(\ds 41 \, 041\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds 11^2\) | \(\nmid\) | \(\ds 41 \, 041\) | |||||||||||
\(\ds 13^2 \times 242 + 143\) | \(=\) | \(\ds 41 \, 041\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds 13^2\) | \(\nmid\) | \(\ds 41 \, 041\) | |||||||||||
\(\ds 41^2 \times 24 + 697\) | \(=\) | \(\ds 41 \, 041\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds 41^2\) | \(\nmid\) | \(\ds 41 \, 041\) |
We also have that:
\(\ds 41 \, 040\) | \(=\) | \(\ds 6840 \times \paren {7 - 1}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 4104 \times \paren {11 - 1}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 3420 \times \paren {13 - 1}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 1026 \times \paren {41 - 1}\) |
The result follows by Korselt's Theorem.
$\blacksquare$