# Definition:Carmichael Number/Examples

Jump to navigation Jump to search

## Examples of Carmichael Numbers

### $561$ is a Carmichael Number

$\forall a \in \Z: a \perp 561: a^{561} \equiv a \pmod {561}$

while $561$ is composite.

### $1105$ is a Carmichael Number

$\forall a \in \Z: a \perp 1105: a^{1105} \equiv a \pmod {1105}$

while $1105$ is composite.

### $1729$ is a Carmichael Number

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

while $1729$ is composite.

### $2465$ is a Carmichael Number

$\forall a \in \Z: a \perp 2465: a^{2465} \equiv a \pmod {2465}$

while $2465$ is composite.

### $41 \, 041$ is a Carmichael Number

$\forall a \in \Z: a \perp 41 \, 041: a^{41 \, 041} \equiv a \pmod {41 \, 041}$

while $41 \, 041$ is composite.

### $294 \, 409$ 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.

### $509 \, 033 \, 161$ 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.