# Euler Lucky Number/Examples/5

## Example of Euler Lucky Number

The expression:

$n^2 + n + 5$

yields primes for $n = 0$ to $n = 3$.

This demonstrates that $5$ is a Euler lucky number.

## Proof

 $\displaystyle 0^2 + 0 + 5$ $=$ $\displaystyle 0 + 0 + 5$ $\displaystyle = 5$ which is prime $\displaystyle 1^2 + 1 + 5$ $=$ $\displaystyle 1 + 1 + 5$ $\displaystyle = 7$ which is prime $\displaystyle 2^2 + 2 + 5$ $=$ $\displaystyle 4 + 2 + 5$ $\displaystyle = 11$ which is prime $\displaystyle 3^2 + 3 + 5$ $=$ $\displaystyle 9 + 3 + 5$ $\displaystyle = 17$ which is prime

$\blacksquare$