Equivalence of Definitions of Euler Lucky Number

Theorem
The following definitions of an Euler lucky number are equivalent:

Proof
Let $p$ be an Euler lucky number.

Let $f_p: \Z \to \Z$ be the mapping defined as:
 * $\forall n \in \Z: f_p \left({n}\right) = n^2 + n + p$

Let $m = n - 1$.

Then:

We have that $f_p \left({n}\right)$ is prime for $0 \le n < p - 1$.

Thus $f_p \left({m}\right)$ is prime for $0 \le \left(n - 1}\right) < p - 1$.

and so $f_p \left({m}\right)$ is prime for $1 \le n < p$.