Euler Lucky Number/Examples/2

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 $2$.

$\blacksquare$