# Euler Lucky Number/Examples/17

## Example of Euler Lucky Number

The expression:

$n^2 + n + 17$

yields primes for $n = 0$ to $n = 15$.

This demonstrates that $17$ is a Euler lucky number.

## Proof

 $\ds 0^2 + 0 + 17$ $=$ $\ds 0 + 0 + 17$ $\ds = 17$ which is prime $\ds 1^2 + 1 + 17$ $=$ $\ds 1 + 1 + 17$ $\ds = 19$ which is prime $\ds 2^2 + 2 + 17$ $=$ $\ds 4 + 2 + 17$ $\ds = 23$ which is prime $\ds 3^2 + 3 + 17$ $=$ $\ds 9 + 3 + 17$ $\ds = 29$ which is prime $\ds 4^2 + 4 + 17$ $=$ $\ds 16 + 4 + 17$ $\ds = 37$ which is prime $\ds 5^2 + 5 + 17$ $=$ $\ds 25 + 5 + 17$ $\ds = 47$ which is prime $\ds 6^2 + 6 + 17$ $=$ $\ds 36 + 6 + 17$ $\ds = 59$ which is prime $\ds 7^2 + 7 + 17$ $=$ $\ds 49 + 7 + 17$ $\ds = 73$ which is prime $\ds 8^2 + 8 + 17$ $=$ $\ds 64 + 8 + 17$ $\ds = 89$ which is prime $\ds 9^2 + 9 + 17$ $=$ $\ds 81 + 9 + 17$ $\ds = 107$ which is prime $\ds 10^2 + 10 + 17$ $=$ $\ds 100 + 10 + 17$ $\ds = 127$ which is prime $\ds 11^2 + 11 + 17$ $=$ $\ds 121 + 11 + 17$ $\ds = 149$ which is prime $\ds 12^2 + 12 + 17$ $=$ $\ds 144 + 12 + 17$ $\ds = 173$ which is prime $\ds 13^2 + 13 + 17$ $=$ $\ds 169 + 13 + 17$ $\ds = 199$ which is prime $\ds 14^2 + 14 + 17$ $=$ $\ds 196 + 14 + 17$ $\ds = 227$ which is prime $\ds 15^2 + 15 + 17$ $=$ $\ds 225 + 15 + 17$ $\ds = 257$ which is prime $\ds 16^2 + 16 + 17$ $=$ $\ds 256 + 16 + 17$ $\ds = 289$ which is not prime, being $17^2$

$\blacksquare$