Euler Lucky Number Function for n equals p

Definition

Let $p$ be a prime number.

Let $f_p: \Z \to \Z$ be the mapping defined as:

$\forall n \in \Z: f_p \left({n}\right) = n^2 - n + p$

Then $f_p \left({p}\right)$ is not prime.

Proof

Let $n = p$.

Then:

\(\ds n^2 - n + p\)

\(=\)

\(\ds p^2- p + p\)

\(\ds \)

\(=\)

\(\ds p^2\)

which is not prime.

$\blacksquare$