Euler Lucky Number/Examples/17

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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).

$\blacksquare$


Sources