Carmichael Number with 4 Prime Factors

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Theorem

$41 \, 041$ is the smallest Carmichael number with $4$ prime factors:

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


Proof

From Sequence of Carmichael Numbers:

The sequence of Carmichael numbers begins:

$561, 1105, 1729, 2465, 2821, 6601, 8911, 10585, 15841, \ldots$


The sequence continues:

$\ldots, 29 \, 341, 41 \, 041$

We now investigate their prime factors:

\(\ds 561\) \(=\) \(\ds 3 \times 11 \times 17\)
\(\ds 1105\) \(=\) \(\ds 5 \times 13 \times 17\)
\(\ds 1729\) \(=\) \(\ds 7 \times 13 \times 19\)
\(\ds 2465\) \(=\) \(\ds 5 \times 17 \times 29\)
\(\ds 2821\) \(=\) \(\ds 7 \times 13 \times 31\)
\(\ds 6601\) \(=\) \(\ds 7 \times 23 \times 41\)
\(\ds 8911\) \(=\) \(\ds 7 \times 19 \times 67\)
\(\ds 10 \, 585\) \(=\) \(\ds 5 \times 29 \times 73\)
\(\ds 15 \, 841\) \(=\) \(\ds 7 \times 31 \times 73\)
\(\ds 29 \, 341\) \(=\) \(\ds 13 \times 37 \times 61\)
\(\ds 41 \, 041\) \(=\) \(\ds 7 \times 11 \times 13 \times 41\)

$\blacksquare$


Sources

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