# Carmichael Number/Examples/41,041

## 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:

 $\displaystyle 7^2 \times 827 + 28$ $=$ $\displaystyle 41 \, 041$ $\displaystyle \leadsto \ \$ $\displaystyle 7^2$ $\nmid$ $\displaystyle 41 \, 041$ $\displaystyle 11^2 \times 339 + 22$ $=$ $\displaystyle 41 \, 041$ $\displaystyle \leadsto \ \$ $\displaystyle 11^2$ $\nmid$ $\displaystyle 41 \, 041$ $\displaystyle 13^2 \times 242 + 143$ $=$ $\displaystyle 41 \, 041$ $\displaystyle \leadsto \ \$ $\displaystyle 13^2$ $\nmid$ $\displaystyle 41 \, 041$ $\displaystyle 41^2 \times 24 + 697$ $=$ $\displaystyle 41 \, 041$ $\displaystyle \leadsto \ \$ $\displaystyle 41^2$ $\nmid$ $\displaystyle 41 \, 041$

We also have that:

 $\displaystyle 41 \, 040$ $=$ $\displaystyle 6840 \times \paren {7 - 1}$ $\displaystyle$ $=$ $\displaystyle 4104 \times \paren {11 - 1}$ $\displaystyle$ $=$ $\displaystyle 3420 \times \paren {13 - 1}$ $\displaystyle$ $=$ $\displaystyle 1026 \times \paren {41 - 1}$

The result follows by Korselt's Theorem.

$\blacksquare$