# Definition:Euler Lucky Number/Definition 2

## Definition

The Euler lucky numbers are the prime numbers $p$ such that:

$n^2 - n + p$

is prime for $1 \le n < p$.

## Examples

### $n^2 + n + 2$

The expression:

$n^2 + n + 2$

yields primes for $n = 0$ and for no other $n \in \Z_{\ge 0}$.

### $n^2 + n + 3$

The expression:

$n^2 + n + 3$

yields primes for $n = 0$ to $n = 1$.

### $n^2 + n + 5$

The expression:

$n^2 + n + 5$

yields primes for $n = 0$ to $n = 3$.

### $n^2 + n + 11$

The expression:

$n^2 + n + 11$

yields primes for $n = 0$ to $n = 9$.

### $n^2 + n + 17$

The expression:

$n^2 + n + 17$

yields primes for $n = 0$ to $n = 15$.

### $n^2 + n + 41$

The expression:

$n^2 + n + 41$

yields primes for $n = 0$ to $n = 39$.

It also generates the same set of primes for $n = -1 \to n = -40$.

These are not the only primes generated by this formula.

No other quadratic function of the form $x^2 + a x + b$, where $a, b \in \Z_{>0}$ and $a, b < 10000$ generates a longer sequence of primes.

## Source of Name

This entry was named for Leonhard Paul Euler.