# Euler Lucky Number/Examples/5

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## 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 |

This sequence is A027690 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).

$\blacksquare$