Equivalence of Definitions of Euler Lucky Number

From ProofWiki
Jump to navigation Jump to search

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:

\(\displaystyle f_p \left({m}\right)\) \(=\) \(\displaystyle f_p \left({n - 1}\right)\)
\(\displaystyle \) \(=\) \(\displaystyle \left({n - 1}\right)^2 + \left({n - 1}\right) + p\)
\(\displaystyle \) \(=\) \(\displaystyle n^2 - 2 n + 1 + n - 1 + p\)
\(\displaystyle \) \(=\) \(\displaystyle n^2 - n + p\)

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 $1 \le n < p$.

$\blacksquare$