Are there infinitely many integer values $n$ such that $q(n)$ is a prime number? Numerical evidence points to a yes answer.
This is similar to Landau's 4th problem from 1912. (The conjecture that there are infinitely many primes $p$ of the form $p=n^2+1 $?) Of course, Landau did not have a computer. Given n a positive number, for what values of $n$ is $q(n)=n^2 + n + 41 $ a prime number? This is known as Prime-Generating Polynomial.
also my document on this quadratic polynomial
There are 3 .pdf files hosted at mersenne.org. I characterize all the cases when n^2 + n + 41 can be a composite number. Assuming positive integer n.
I show a data table, graph, and curve fit to characterize all the cases when this trinomial is a composite number, up to a certain numerical limit.
Also, I have found some algebraic factorizations for q(n)
A leading question is, "If we can know whenever n^2+n+41 is composite, what does that tell up about when that trinomial is a prime number?"
Let me know if there are any questions.