Euler Lucky Number/Examples/11

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

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