Lucas-Carmichael Number/Examples/2015

Example of Lucas-Carmichael Number

$2015$ is a Lucas-Carmichael number:

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

Proof

We have that:

$2015 = 5 \times 13 \times 31$

Then:

\(\ds 2016\)

\(=\)

\(\ds 6 \times 336\)

so $\paren {5 + 1} \divides \paren {2015 + 1}$

\(\ds \)

\(=\)

\(\ds 14 \times 144\)

so $\paren {13 + 1} \divides \paren {2015 + 1}$

\(\ds \)

\(=\)

\(\ds 32 \times 63\)

so $\paren {31 + 1} \divides \paren {2015 + 1}$

$\blacksquare$