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.

see link https://mathworld.wolfram.com/Prime-GeneratingPolynomial.html

also Wikipedia

https://en.wikipedia.org/wiki/Formula_for_primes#Prime_formulas_and_polynomial_functions

also my document on this quadratic polynomial

https://mersenneforum.org/showthread.php?p=581027#post581027

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)

https://sites.google.com/site/mattc1anderson/prime-producing-polynomial

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.

Matt

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