# What are prime number values of the trinomial $q(n) = n^2 + n + 41$? Assuming $n$ is a positive integer

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 Wikipedia

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

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.

Matt

• This is an open problem. Jun 15, 2021 at 9:17
• How can numerical evidence point to something in this case? Jun 15, 2021 at 13:15
• The Bunyakovsky conjecture en.wikipedia.org/wiki/Bunyakovsky_conjecture says that, if $f(n)$ is an irreducible polynomial with positive leading term and there is no modulus $M$ for which $f(n)$ is identically $0 \bmod M$, then $f(n)$ is prime infinitely often. The polynomial $n^2+n+41$ satisfies these criteria. But there is no polynomial of degree $\geq 2$ for which the Bunyakovsky conjecture has been proved. Jun 15, 2021 at 13:44
• @MarkSapir One can make a conjectural asymptotic for the number of $n< X$ with $n^2+n+41$ prime, find numerical evidence that this asymptotic holds, and observe that this asymptotic predicts that there are infinitely many prime values. Jun 21, 2021 at 0:37
• Furthermore the method I suggested in this case, of first make a prediction based on the best available heuristics, then use numerics to check how well the heuristics seem to hold in this case, would not run into trouble in the 4n+1 vs. 4n+3 case anyways. Jun 28, 2021 at 18:40

Since the ring of integers of $$\mathbb{Q}[\sqrt{-163}]$$ is a PID, it follows that a rational prime $$p \neq 163$$ may be expressed in the form $$x^{2} + xy + 41y^{2}$$ for rational integers $$x$$ and $$y$$ if and only if $$p$$ is a quadratic residue (mod $$163$$).(This is well-known). But, as you point out yourself, your question is comparable to asking how many primes $$p$$ have the form $$n^{2} +1$$ for integer $$n$$, which is well known to be open and Wojowu confirms in comments that your question is open too.
Later edit: I find it mildly interesting that the prime $$p$$ is expressible in this way (ie $$p = n^{2}+n+41$$) if and only if $$p$$ is expressible as the sum of four integer squares in one of the following ways: If $$n$$ is odd, we find that $$p = \left( \frac{n-9}{2} \right)^{2} + \left( \frac{n+1}{2} \right)^{2} +\left( \frac{n+1}{2} \right)^{2} + \left( \frac{n+9}{2} \right)^{2}$$ and if $$n$$ is even we find that $$p = \left( \frac{n-8}{2} \right)^{2} + \left( \frac{n}{2} \right)^{2} +\left( \frac{n}{2} \right)^{2} + \left( \frac{n+10}{2} \right)^{2}.$$
Even later edit: For any prime $$p \neq 41$$ which is a quadratic residue (mod $$163$$), there is a unique integer $$h$$ with $$1 \leq h \leq \frac{p-1}{2}$$ such that $$p$$ divides $$h^{2}+h+41$$, and then $$p$$ is necessarily the largest prime divisor of $$h^{2}+h+41.$$ An inductive argument of a type which dates back to Euler and/or Fermat then shows that $$p$$ is necessarily of the form $$x^{2}+xy+41y^{2}$$ for integers $$x$$ and $$y$$, and allows you to explicitly determine $$x$$ and $$y$$, given such an expression for the other (smaller) prime divisors $$q$$ of $$h^{2}+h+41$$ (all of which are also necessarily quadratic residues (mod $$163$$)).