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:
\(\ds 509 \, 033 \, 160\) | \(=\) | \(\ds 2^3 \times 3^4 \times 5 \times 157 \, 109\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 84 \, 838 \, 860 \times 6\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 42 \, 419 \, 430 \times 12\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 28 \, 279 \, 620 \times 18\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 14 \, 139 \, 810 \times 36\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 7 \, 069 \, 905 \times 72\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 4 \, 713 \, 270 \times 108\) |
Thus $509 \, 033 \, 161$ is a Carmichael number by Korselt's Theorem.
Then we have:
and:
$\blacksquare$
Sources
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $509,033,161$