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:
\(\ds 294 \, 408\) | \(=\) | \(\ds 2^3 \times 3^3 \times 29 \times 47\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 8178 \times 36\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 4089 \times 72\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 2726 \times 108\) |
Thus $294 \, 409$ is a Carmichael number by Korselt's Theorem.
$\blacksquare$
Sources
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $509,033,161$