Euler Lucky Number/Examples/41

Example of Polynomial Expressions for Primes
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.

Proof
Then we have:

and so replacing $0$ to $39$ with $-1$ to $-40$ yields exactly the same sequence of primes.

We note in addition the example:
 * $581^2 + 581 + 41 = 338 \, 183$

which is prime.