# 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 $509 \, 033 \, 161$:

$\forall p \divides 509 \, 033 \, 161: p^2 \nmid 509 \, 033 \, 161$

We also see that:

 $\ds 509 \, 033 \, 160$ $=$ $\ds 2^3 \times 3^4 \times 5 \times 157 \, 109$ $\ds$ $=$ $\ds 84 \, 838 \, 860 \times 6$ $\ds$ $=$ $\ds 42 \, 419 \, 430 \times 12$ $\ds$ $=$ $\ds 28 \, 279 \, 620 \times 18$ $\ds$ $=$ $\ds 14 \, 139 \, 810 \times 36$ $\ds$ $=$ $\ds 7 \, 069 \, 905 \times 72$ $\ds$ $=$ $\ds 4 \, 713 \, 270 \times 108$

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.

$\blacksquare$