# Carmichael Number/Examples/294,409

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## 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.

$\blacksquare$

## Sources

- 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $509,033,161$