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 $17^2$ |
This sequence is A007635 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).
$\blacksquare$