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2021-06-15, 09:08   #1
MattcAnderson

"Matthew Anderson"
Dec 2010
Oregon, USA

3×172 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
Attached Files
 A prime producing polynomial March 9 2021.pdf (269.4 KB, 46 views) A prime producing quadratic expression 2019 (3).pdf (473.3 KB, 29 views)

Last fiddled with by MattcAnderson on 2021-06-26 at 09:19 Reason: added slideshow file, changed trinomial name from q to f.

2021-06-16, 07:42   #2
MattcAnderson

"Matthew Anderson"
Dec 2010
Oregon, USA

3×172 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
Attached Files
 small equation coefficient doublecheck 33.pdf (631.8 KB, 24 views)

Last fiddled with by MattcAnderson on 2021-06-16 at 13:35 Reason: explained method

2021-08-15, 21:38   #3
MattcAnderson

"Matthew Anderson"
Dec 2010
Oregon, USA

3×172 Posts
project to date

Hi again all,

Here is a 4 page write-up with all the important points to date.

Regards,

Matt C Anderson
Attached Files
 Prime Producing Polynomial August 2021.pdf (286.6 KB, 11 views)

2021-09-03, 13:47   #4
MattcAnderson

"Matthew Anderson"
Dec 2010
Oregon, USA

3·172 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
Attached Files
 count of prime values of n^2+n+41.pdf (116.5 KB, 5 views)

 2021-09-03, 13:51 #5 MattcAnderson     "Matthew Anderson" Dec 2010 Oregon, USA 3·172 Posts Hi All, I asked about this polynomial x^2+x+41 on mathoverflow.net see https://mathoverflow.net/questions/3...-41-assuming-n Regards, Matt
 2021-09-03, 16:14 #6 Dr Sardonicus     Feb 2017 Nowhere 22·3·401 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) = x2 + 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.
 2021-09-04, 14:22 #7 sweety439     "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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