Euler Lucky Number/Examples/11

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Example of Euler Lucky Number

The expression:

$n^2 + n + 11$

yields primes for $n = 0$ to $n = 9$.


This demonstrates that $11$ is a Euler lucky number.


Proof

\(\ds 0^2 + 0 + 11\) \(=\) \(\ds 0 + 0 + 11\) \(\ds = 11\) which is prime
\(\ds 1^2 + 1 + 11\) \(=\) \(\ds 1 + 1 + 11\) \(\ds = 13\) which is prime
\(\ds 2^2 + 2 + 11\) \(=\) \(\ds 4 + 2 + 11\) \(\ds = 17\) which is prime
\(\ds 3^2 + 3 + 11\) \(=\) \(\ds 9 + 3 + 11\) \(\ds = 23\) which is prime
\(\ds 4^2 + 4 + 11\) \(=\) \(\ds 16 + 4 + 11\) \(\ds = 31\) which is prime
\(\ds 5^2 + 5 + 11\) \(=\) \(\ds 25 + 5 + 11\) \(\ds = 41\) which is prime
\(\ds 6^2 + 6 + 11\) \(=\) \(\ds 36 + 6 + 11\) \(\ds = 53\) which is prime
\(\ds 7^2 + 7 + 11\) \(=\) \(\ds 49 + 7 + 11\) \(\ds = 67\) which is prime
\(\ds 8^2 + 8 + 11\) \(=\) \(\ds 64 + 8 + 11\) \(\ds = 83\) which is prime
\(\ds 9^2 + 9 + 11\) \(=\) \(\ds 81 + 9 + 11\) \(\ds = 101\) which is prime

This sequence is A048058 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).

$\blacksquare$