# Carmichael Number/Examples/294,409

## Example of Carmichael Number

$294 \, 409$ is a Carmichael number:

$\forall a \in \Z: a \perp 294 \, 409: a^{294 \, 409} \equiv a \pmod {294 \, 409}$

while $294 \, 409$ is composite.

## Proof

We have that:

$294 \, 409 = 37 \times 73 \times 109$

First note that $294 \, 409$ is square-free.

Hence the square of none of its prime factors is a divisor of $294 \, 409$:

$\forall p \divides 294 \, 409: p^2 \nmid 294 \, 409$

We also see that:

 $\displaystyle 294 \, 408$ $=$ $\displaystyle 2^3 \times 3^3 \times 29 \times 47$ $\displaystyle$ $=$ $\displaystyle 8178 \times 36$ $\displaystyle$ $=$ $\displaystyle 4089 \times 72$ $\displaystyle$ $=$ $\displaystyle 2726 \times 108$

Thus $294 \, 409$ is a Carmichael number by Korselt's Theorem.

$\blacksquare$