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Consider the set $\mathcal{E}:=\{n\in\mathbb{N} \text{ such that } n^2+n+41 \text{ is prime}\}$ (this is A056561 on the OEIS).

Now, after some little trial and error, it appears to be well fitted by the graph of $g$ defined by $g(k):=0.1468k+0.2773k^{1.0667}\log(k)^{0.729}$ (where $k\in\mathbb{N}$) in the sense that the following seems to hold:

$\forall n\in\mathcal{E}\cap[\![0;27400]\!], \exists k\in [\![0;10000]\!] \text{ such that } n\in[g(k)-200;g(k)+200]\ \ (*)$

Here is a plot illustrating this: enter image description here

This seems to be related to the first Hardy-Littlewood conjecture, see the answer of Joriki in this older question, but not quite in the same way as in that answer.

It also has a bit of the flavor of Polymath8b on the existence of bounded intervals with many consecutive primes.

[edit of august 19: it could also be just a consequence of the PNT, which asserts in particular that the $m-$th prime is $p_m\simeq m\log(m)$, so that for any $n\in\mathcal{E}$ there exist some $m$ such that $n^2+n+41=p_m$, so $n\simeq \frac{1}{2}(\sqrt{4p_m-163}-1)\simeq m^{0.5}\log(m)^{0.5}$. But these are not quite the constants in $g$, perhaps due to its additional linear term.]

As such, it might follow from known results, or at least widely believed conjectures, but unfortunately I am not knowledgeable enough to be quite sure.

  1. Can $(*)$ indeed be proved for any $n\in\mathcal{E}$ (i.e. still with a fixed finite amplitude around $g(k)$, perhaps a bit larger than 400) with current technology? Or at least assuming some HL type conjectures?

  2. If yes, what would be an executive-style summary of the proof?

(Just for reference, another related question on MSE, but with the polynomial $n^2-n+41$ instead and about values for which it is not prime, is this one).

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Thomas Sauvaget
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  • Do you mean $g$ as defined with exactly those constants? It is extremely unlikely, just because the asymptotic density of numbers within $400$ of $g(\mathbb N)$ is $0$. – Robert Israel Aug 18 '17 at 22:19
  • Thank you for the comment. As I mentioned, perhaps the 400 should be increased somewhat. That being said, I do not quite follow why your comment would make it unlikely. If I understand it well, you are saying that as $k$ grows large, the integers within distance 400 of $g(k)$ represent less and less of the total amount of integers in $[0;g(k)+400]$. If so, while true, how would that contradict the guess that for any fixed $k$, however large, among the about 400 integers around $g(k)$ one of them will be the next element of $\mathcal{E}$ (i.e. that $g(k)$ is always a good approximation) ? – Thomas Sauvaget Aug 19 '17 at 00:07

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