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