# Definition:Euler Lucky Number

## Definition

### 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 $1 \le n < p$.

## Sequence of Euler Lucky Numbers

The complete sequence of Euler lucky numbers is:

$2, 3, 5, 11, 17, 41$

## 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.

## Historical Note

When Charles Babbage built a trial version of his analytical engine, he set it to work calculating a list of values of Euler lucky numbers.

David Brewster reported:

thirty-two numbers of the same table were calculated in the space of two minutes and thirty seconds; and as these contained eighty-two figures, the engine produced thirty-three figures every minute, or more than one figure in every two seconds. On another occasion it produced forty-four figures per minute. This rate of computation could be maintained for any length of time; and it is probable that few writers are able to copy with equal speed for many hours together.