Who found the expression $n^2 - n + 41 $ for generating prime numbers?

Question

I am doing some research and I cannot seem to find the answer anywhere so does anyone know who found the expression $n^2 - n + 41 $ for generating prime numbers?

Answer 1

It's Euler who first found it. He mentions that formula in 1771 in a letter to Bernoulli. The relevant passage is shown below, in particular the second paragraph:

My translation:

$\qquad$ The biggest prime number that we know of is without doubt $\mathfrak{2^{31}-1=2137483647}$, which Fermat assured to be prime, $\mathfrak{\&}$ I also proved that; because this formula will never admit other divisors other than one $\mathfrak{\&}$ or the other of these $\mathfrak{2}$ forms $\mathfrak{248n+1\ \& \ 248n+63}$, I have examined all prime numbers contained in these two formulas until $\mathfrak{46339}$, and none was found to be a divisor.

$\qquad $ This progression $\mathfrak{41.\ 43.\ 47.\ 53.\ 61.\ 71.\ 83.\ 97.\ 113.\ 131\ \, \& \rm c.}$ whose general term is $\mathfrak{41+ }x\mathfrak{+}xx$, is as much remarkable since the $\mathfrak{40}$ first terms are all prime numbers.

Answer 2

Euler first noticed (in 1772) that the quadratic polynomial $$ P(n) = n^2 + n + 41 $$ is prime for all natural numbers less than 40.

from Wiki:Formula for primes

Answer 3

As others have noted, Euler, in 1772 published this result in a very similar form (see @draks... answer).

Note however, that he published the quadratic $n^2+n+41$, not $n^2-n+41$.

My contribution: This result was stated in Nouveaux Mémoires de l'Académie royale des Sciences. Berlin, p. 36, 1772, by L. Euler.

Answer 4

Some context, the polynomial $n^2 + n + 41$ is prime for $0 \leq n \leq 39.$ It cannot possibly be prime for $n=40$ or $n=41$ because $n^2 + n + 41$ is then divisible by $41$ but larger than $41.$

It is a simple result, that $n^2 + n + k$ can only represent such a large initial sequence of primes, $0 \leq n \leq k-2,$ when both $k$ and $4k-1$ are prime. Furthermore, Rabinowitz, 1913, given those conditions, showed this happens if and only if the class number of discriminant $-(4k-1)$ is one. I give a proof of this at Is the notorious $n^2 + n + 41$ prime generator the last of its type?

Later, https://en.wikipedia.org/wiki/Stark%E2%80%93Heegner_theorem this was shown to be the last time this happens.