# Euler Lucky Number/Examples/2

## Example of Euler Lucky Number

The expression:

$n^2 + n + 2$

yields primes for $n = 0$ and for no other $n \in \Z_{\ge 0}$.

This demonstrates that $2$ (trivially) is a Euler lucky number.

## Proof

 $\displaystyle 0^2 + 0 + 2$ $=$ $\displaystyle 0 + 0 + 2$ $\displaystyle = 2$ which is prime

Let $n > 0$.

Then:

 $\displaystyle n^2 + n + 2$ $=$ $\displaystyle n \left({n + 1}\right) + 2$ $\displaystyle$ $=$ $\displaystyle 2 \left({\dfrac {n \left({n + 1}\right)} 2}\right) + 2$ $\displaystyle$ $=$ $\displaystyle 2 \left({\dfrac {n \left({n + 1}\right)} 2 + 1}\right)$

and so is divisible by $2$.

$\blacksquare$