Carmichael Number/Examples/509,033,161

Example of Carmichael Number
$509 \, 033 \, 161$ is a Carmichael number:
 * $\forall a \in \Z: a \perp 509 \, 033 \, 161: a^{509 \, 033 \, 161} \equiv a \pmod {509 \, 033 \, 161}$

while $509 \, 033 \, 161$ is composite.

Also:
 * $509 \, 033 \, 161 = 1729 \times 294 \, 409$

while both $1729$ and $294 \, 409$ are themselves Carmichael numbers.

Proof
We have that:
 * $509 \, 033 \, 161 = 7 \times 13 \times 19 \times 37 \times 73 \times 109$

First note that $509 \, 033 \, 161$ is square-free.

Hence the square of none of its prime factors is a divisor of $509 \, 033 \, 161$:


 * $\forall p \mathrel \backslash 509 \, 033 \, 161: p^2 \nmid 509 \, 033 \, 161$

We also see that:

Thus $509 \, 033 \, 161$ is a Carmichael number by Korselt's Theorem.

Then we have:
 * $1729$ is a Carmichael number

and:
 * $294 \, 409$ is a Carmichael number.