Question

Show that 2821 is a Carmichael number.

Short answer

2821 is a Carmichael number.

Step-by-step solution

Step: 1

We must show that $b^{2820}\equiv 1(mod2821)$for all b relatively prime to 2821.

Note that 2821=7.13.31, and if $gcd(b,2821)=1$, then

$\gcd (b,7)=\gcd (b,13)=\gcd (b,31)=1$

.

Using Fermat’s little theorem we find that

$\begin{array}{r}{b}^6\equiv 1(mod7)\\ b^{12}\equiv 1(mod13)\\ b^{30}\equiv 1(mod31)\end{array}$

Step: 2

It follows that

$\begin{array}{r}{b}^{2820}={\left(b^6\right)}^{470}=1(mod7)\\ b^{2820}\equiv {\left(b^{12}\right)}^{235}\equiv 1(mod13)\\ b^{2820}\equiv {\left(b^{30}\right)}^{94}\equiv 1(mod31)\end{array}$

Step: 3

By Chinese remainder theorem, it follows that $b^{2820}\equiv 1(mod2821)$as desired.