Carmichael Number/Examples/41,041

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Example of Carmichael Number

$41 \, 041$ is a Carmichael number:

$\forall a \in \Z: a \perp 41 \, 041: a^{41 \, 041} \equiv a \pmod {41 \, 041}$

while $41 \, 041$ is composite.


Proof

We have that:

$41,041 = 7 \times 11 \times 13 \times 41$

and so:

\(\displaystyle 7^2 \times 827 + 28\) \(=\) \(\displaystyle 41 \, 041\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle 7^2\) \(\nmid\) \(\displaystyle 41 \, 041\)
\(\displaystyle 11^2 \times 339 + 22\) \(=\) \(\displaystyle 41 \, 041\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle 11^2\) \(\nmid\) \(\displaystyle 41 \, 041\)
\(\displaystyle 13^2 \times 242 + 143\) \(=\) \(\displaystyle 41 \, 041\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle 13^2\) \(\nmid\) \(\displaystyle 41 \, 041\)
\(\displaystyle 41^2 \times 24 + 697\) \(=\) \(\displaystyle 41 \, 041\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle 41^2\) \(\nmid\) \(\displaystyle 41 \, 041\)


We also have that:

\(\displaystyle 41 \, 040\) \(=\) \(\displaystyle 6840 \times \paren {7 - 1}\)
\(\displaystyle \) \(=\) \(\displaystyle 4104 \times \paren {11 - 1}\)
\(\displaystyle \) \(=\) \(\displaystyle 3420 \times \paren {13 - 1}\)
\(\displaystyle \) \(=\) \(\displaystyle 1026 \times \paren {41 - 1}\)

The result follows by Korselt's Theorem.

$\blacksquare$