The Prime Producing Polynomial

Starting observation

$f(n) = n^2 + n + 41$

Feed in $n = 0, 1, 2, \dots, 39$ and out comes a prime number every single time. Forty primes in a row. Then at $n = 40$ the spell breaks: $f(40) = 40 \times 41 + 41 = 41^2$, which is not prime. Don't read this collapse as a disappointment. The collapse is the doorway.

The research arc

A research question is never settled in one sitting. It moves through four stages — and a notebook with an AI assistant can write the code, so nobody is blocked by programming:

Query → Data → Pattern → Abstraction

A model chain of questions

1. Replace $41$ by $42$. Does $n^2 + n + 42$ produce many primes? What goes wrong when the constant is composite?
2. So suppose the constant $k$ is prime: $43, 47, 53$, and so on. Notice that $n = k$ and $n = k - 1$ always fail. So look at $n$ from $0$ to $k - 2$: how many outputs are prime?
3. Define the prime density of $k$: the number of prime outputs for $n = 0$ to $k - 2$, divided by the number of inputs. Compute it for many primes $k$. Plot it. What is the pattern? Which $k$ turn out unusually lucky, and can you guess why?

Your task

Compose one open-ended question of your own, starting from this polynomial or from any pattern you noticed in it. Creating the question is the crucial step of the whole research process — and it is the step teachers are almost never asked to practise. Then sketch how a group of students would carry your question through the four stages of the arc.