Carmichael Number/Examples/509,033,161

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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:

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

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.

$\blacksquare$


Sources