Equivalence of Definitions of Euler Lucky Number
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Theorem
The following definitions of the concept of Euler Lucky Number are equivalent:
Definition 1
The Euler lucky numbers are the prime numbers $p$ such that:
- $n^2 + n + p$
is prime for $0 \le n < p - 1$.
Definition 2
The Euler lucky numbers are the prime numbers $p$ such that:
- $n^2 - n + p$
is prime for $1 \le n < p$.
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:
\(\ds f_p \left({m}\right)\) | \(=\) | \(\ds f_p \left({n - 1}\right)\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \left({n - 1}\right)^2 + \left({n - 1}\right) + p\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds n^2 - 2 n + 1 + n - 1 + p\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds n^2 - n + p\) |
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 $1 \le n < p$.
$\blacksquare$