mersenneforum.org  

Go Back   mersenneforum.org > Extra Stuff > Blogorrhea > MattcAnderson

Reply
 
Thread Tools
Old 2021-06-15, 09:08   #1
MattcAnderson
 
MattcAnderson's Avatar
 
"Matthew Anderson"
Dec 2010
Oregon, USA

3×172 Posts
Default 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 2021-06-26 at 09:19 Reason: added slideshow file, changed trinomial name from q to f.
MattcAnderson is offline   Reply With Quote
Old 2021-06-16, 07:42   #2
MattcAnderson
 
MattcAnderson's Avatar
 
"Matthew Anderson"
Dec 2010
Oregon, USA

3×172 Posts
Default

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
File Type: pdf 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
MattcAnderson is offline   Reply With Quote
Old 2021-08-15, 21:38   #3
MattcAnderson
 
MattcAnderson's Avatar
 
"Matthew Anderson"
Dec 2010
Oregon, USA

3×172 Posts
Smile 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
File Type: pdf Prime Producing Polynomial August 2021.pdf (286.6 KB, 11 views)
MattcAnderson is offline   Reply With Quote
Old 2021-09-03, 13:47   #4
MattcAnderson
 
MattcAnderson's Avatar
 
"Matthew Anderson"
Dec 2010
Oregon, USA

3·172 Posts
Default

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
File Type: pdf count of prime values of n^2+n+41.pdf (116.5 KB, 5 views)
MattcAnderson is offline   Reply With Quote
Old 2021-09-03, 13:51   #5
MattcAnderson
 
MattcAnderson's Avatar
 
"Matthew Anderson"
Dec 2010
Oregon, USA

3·172 Posts
Smile

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
MattcAnderson is offline   Reply With Quote
Old 2021-09-03, 16:14   #6
Dr Sardonicus
 
Dr Sardonicus's Avatar
 
Feb 2017
Nowhere

22·3·401 Posts
Default

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.
Dr Sardonicus is offline   Reply With Quote
Old 2021-09-04, 14:22   #7
sweety439
 
sweety439's Avatar
 
"99(4^34019)99 palind"
Nov 2016
(P^81993)SZ base 36

2·17·89 Posts
Default

What is the natural density of A056561? Is it zero? Or positive?
sweety439 is offline   Reply With Quote
Reply

Thread Tools


Similar Threads
Thread Thread Starter Forum Replies Last Post
prime producing polynomial 2 MattcAnderson MattcAnderson 32 2021-08-15 21:35
Polynomial Discriminant is n^k for an n-1 degree polynomial carpetpool Miscellaneous Math 14 2017-02-18 19:46
Composite Producing Polynomial Project MattcAnderson MattcAnderson 4 2016-09-06 14:15
prime producing polynomial x^2 + x + 41 MattcAnderson MattcAnderson 28 2014-04-26 01:35
Prime Producing Polynomial MattcAnderson MattcAnderson 0 2013-07-19 04:25

All times are UTC. The time now is 05:20.


Mon Sep 20 05:20:46 UTC 2021 up 58 days, 23:49, 0 users, load averages: 1.20, 1.38, 1.35

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.