Euler Lucky Number/Examples/2
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Example of Euler Lucky Number
The expression:
- $n^2 + n + 2$
yields primes for $n = 0$ and for no other $n \in \Z_{\ge 0}$.
This demonstrates that $2$ (trivially) is a Euler lucky number.
Proof
\(\ds 0^2 + 0 + 2\) | \(=\) | \(\ds 0 + 0 + 2\) | \(\ds = 2\) | which is prime |
Let $n > 0$.
Then:
\(\ds n^2 + n + 2\) | \(=\) | \(\ds n \left({n + 1}\right) + 2\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 2 \left({\dfrac {n \left({n + 1}\right)} 2}\right) + 2\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 2 \left({\dfrac {n \left({n + 1}\right)} 2 + 1}\right)\) |
and so is divisible by $2$.
$\blacksquare$