Definition:Lucas-Carmichael Number

Definition

Let $n \in \Z_{>0}$ be a positive composite number which is odd and square-free.

Let $n$ have the property that:

$p \divides n \implies \paren {p + 1} \divides \paren {n + 1}$

where:

$p$ is prime

$p \divides n$ denotes that $p$ is a divisor of $n$.

Then $n$ is classified as a Lucas-Carmichael number.

Sequence

The sequence of Lucas-Carmichael numbers begins:

$399, 935, 2015, 2915, 4991, 5719, 7055, 8855, 12719, 18095, 20705, \ldots$

Examples

399 is a Lucas-Carmichael Number

$399$ is a Lucas-Carmichael number:

$p \divides 399 \implies \paren {p + 1} \divides 400$

935 is a Lucas-Carmichael Number

$935$ is a Lucas-Carmichael number:

$p \divides 935 \implies \paren {p + 1} \divides 936$

2015 is a Lucas-Carmichael Number

$2015$ is a Lucas-Carmichael number:

$p \divides 2015 \implies \paren {p + 1} \divides 2016$

2915 is a Lucas-Carmichael Number

$2915$ is a Lucas-Carmichael number:

$p \divides 2915 \implies \paren {p + 1} \divides 2916$

Source of Name

This entry was named for François Édouard Anatole Lucas and Robert Daniel Carmichael.

Sources

• 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $399$