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:

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