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 $1 \le n < p$.
Also see
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.
Sources
- Weisstein, Eric W. "Lucky Number of Euler." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/LuckyNumberofEuler.html