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

\(\displaystyle 0^2 + 0 + 2\) \(=\) \(\displaystyle 0 + 0 + 2\) \(\displaystyle = 2\) which is prime


Let $n > 0$.

Then:

\(\displaystyle n^2 + n + 2\) \(=\) \(\displaystyle n \left({n + 1}\right) + 2\)
\(\displaystyle \) \(=\) \(\displaystyle 2 \left({\dfrac {n \left({n + 1}\right)} 2}\right) + 2\)
\(\displaystyle \) \(=\) \(\displaystyle 2 \left({\dfrac {n \left({n + 1}\right)} 2 + 1}\right)\)

and so is divisible by $2$.

$\blacksquare$