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.
Also see
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.