Carmichael Number/Examples/1729

Example of Carmichael Number

$1729$ is a Carmichael number:

$\forall a \in \Z: a \perp 1729: a^{1729} \equiv a \pmod {1729}$

while $1729$ is composite.

Proof

We have that:

$1729 = 7 \times 13 \times 19$

and so:

\(\ds 7^2\)

\(\nmid\)

\(\ds 1729\)

\(\ds 13^2\)

\(\nmid\)

\(\ds 1729\)

\(\ds 19^2\)

\(\nmid\)

\(\ds 1729\)

We also have that:

\(\ds 1728\)

\(=\)

\(\ds 288 \times 6\)

\(\ds \)

\(=\)

\(\ds 144 \times 12\)

\(\ds \)

\(=\)

\(\ds 96 \times 18\)

The result follows by Korselt's Theorem.

$\blacksquare$

Sources

• 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $509,033,161$