20210615, 09:08  #1 
"Matthew Anderson"
Dec 2010
Oregon, USA
5×181 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
1611_{8} 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
5×181 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
5×181 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
1611_{8} 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
2×47×53 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
2×3×509 Posts 
What is the natural density of A056561? Is it zero? Or positive?

20210923, 14:05  #8 
"Matthew Anderson"
Dec 2010
Oregon, USA
5·181 Posts 
Hi sweety439 and all,
I learned a new phrase, "natural density" from Wikipedia. It is an open problem whether or not f(n) = n^2+n+41 is a prime number for an infinite number of positive integers n. If f(n) is prime only a finite number of times, then the natural density of f(n) as n goes to infinity would be 0. Also, it is possible, that even if f(n) is prime an infinite number of times, the natural density could still be 0. I wrote this Maple Code, and found some data points. > # A056561 from OEIS.org Numbers n such that n^2+n+41 is prime. > # n^2+n+41 is a prime number for 0<=n<=39. > > count := 0; > for n to 1000 do if isprime(n^2+n+41) then count := count+1; print("n making n^2+n+41 prime", n, "natural density", evalf(count/n)) end if; end do; Let f(n) = n^2+n+41. What is the natural density of f as a gets large? Assume 'n' is a nonnegative integer. My data from Maple calculations  n Natural density 39 1 100 0.86 1,000 0.58 10,000 0.41 My guess is that the natural density is greater than zero. Regards, Matt 
20210923, 15:46  #9  
Feb 2017
Nowhere
11566_{8} Posts 
Quote:
If N > 163 the density of n for which n^{2} + n + 41 is not divisible by any prime < N is . where is the quadratic character of p (mod 163). This accounts for half the primes. It can be shown that the product tends to 0 as N increases without bound. Last fiddled with by Dr Sardonicus on 20210923 at 18:58 Reason: fignix topsy 

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