# Carmichael Number/Examples/1729

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## Example of Carmichael Number

$1729$ is a Carmichael number:

- $\forall a \in \Z: a \perp 1729: a^{1729} \equiv a \pmod {1729}$

while $1729$ is composite.

## Proof

We have that:

- $1729 = 7 \times 13 \times 19$

and so:

\(\displaystyle 7^2\) | \(\nmid\) | \(\displaystyle 1729\) | |||||||||||

\(\displaystyle 13^2\) | \(\nmid\) | \(\displaystyle 1729\) | |||||||||||

\(\displaystyle 19^2\) | \(\nmid\) | \(\displaystyle 1729\) |

We also have that:

\(\displaystyle 1728\) | \(=\) | \(\displaystyle 288 \times 6\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle 144 \times 12\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle 96 \times 18\) |

The result follows by Korselt's Theorem.

$\blacksquare$

## Sources

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