# 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.

$\blacksquare$