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

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

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

$\blacksquare$