20210615, 09:08  #1 
"Matthew Anderson"
Dec 2010
Oregon, USA
3×17^{2} Posts 
prime producing polynomial
See this polynomial
f(n) = n^2 + n + 41 assume n is a positive integer I once received a standing ovation for a presentation on this topic at a 3 day math conference. It was at Salishan Oregon USA, at a conference for community college math teachers. I have done some community college math teaching. I hope you find this interesting. Regards, Matt Last fiddled with by MattcAnderson on 20210626 at 09:19 Reason: added slideshow file, changed trinomial name from q to f. 
20210616, 07:42  #2 
"Matthew Anderson"
Dec 2010
Oregon, USA
3×17^{2} Posts 
Here is a list of many algebraic factorization s to find cases when
f(n) = n^2 + n + 41 is a composite number. I used a data table from a Maple calculation to list numbers when f(n) is a composite number. Then I used the method of 3 point quadratic curve fit to list parabolas. The parabolas are parametric and for all integers on these parabolic curves, f(n) is a composite number. (There are no graphs in this file.) look Matt Last fiddled with by MattcAnderson on 20210616 at 13:35 Reason: explained method 
20210815, 21:38  #3 
"Matthew Anderson"
Dec 2010
Oregon, USA
3×17^{2} Posts 
project to date
Hi again all,
Here is a 4 page writeup with all the important points to date. Regards, Matt C Anderson 
20210903, 13:47  #4 
"Matthew Anderson"
Dec 2010
Oregon, USA
3·17^{2} Posts 
Hi All,
Here is some numerical evidence that there are infinitely many x such that x^2+x+41 is a prime number. See the attached graph. Regards, Matt 
20210903, 13:51  #5 
"Matthew Anderson"
Dec 2010
Oregon, USA
3·17^{2} Posts 
Hi All,
I asked about this polynomial x^2+x+41 on mathoverflow.net see https://mathoverflow.net/questions/3...41assumingn Regards, Matt 
20210903, 16:14  #6 
Feb 2017
Nowhere
2^{2}·3·401 Posts 
Responses at MathOverflow cover most of the ground. In particuar, numerical evidence doesn't address questions of infinitude.
One point  related to one of the responses  is that p is a prime factor of f(x) = x^{2} + x + 41 for some positive integer x when f(x) (mod p) splits into linear factors. This means that the discriminant 163 is a quadratic residue (mod p), which means [thanks to quadratic reciprocity!] that p is a quadratic residue (mod 163). The smallest prime p which is a quadratic residue (mod 163) is p = 41. Thus, f(x), x positive integer, is never divisible by any prime less than 41. 
20210904, 14:22  #7 
"99(4^34019)99 palind"
Nov 2016
(P^81993)SZ base 36
2·17·89 Posts 
What is the natural density of A056561? Is it zero? Or positive?

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