mersenneforum.org prime producing polynomial
 User Name Remember Me? Password
 Register FAQ Search Today's Posts Mark Forums Read

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?

 Thread Tools

 Similar Threads Thread Thread Starter Forum Replies Last Post MattcAnderson MattcAnderson 32 2021-08-15 21:35 carpetpool Miscellaneous Math 14 2017-02-18 19:46 MattcAnderson MattcAnderson 4 2016-09-06 14:15 MattcAnderson MattcAnderson 28 2014-04-26 01:35 MattcAnderson MattcAnderson 0 2013-07-19 04:25

All times are UTC. The time now is 06:08.

Mon Sep 20 06:08:52 UTC 2021 up 59 days, 37 mins, 0 users, load averages: 2.86, 2.03, 1.50

Powered by vBulletin® Version 3.8.11
Copyright ©2000 - 2021, Jelsoft Enterprises Ltd.

This forum has received and complied with 0 (zero) government requests for information.

Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation.
A copy of the license is included in the FAQ.