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2016-03-23, 17:06   #23
chris2be8

Sep 2009

3·659 Posts

Quote:
 Originally Posted by CRGreathouse This is a good question. Let f(x) be the putative prime-producing polynomial. gcd(x,1) = 1 for all x, so f(1) = p is prime. But then f(p + 1), f(2p + 1), ..., are all divisible by p, since each power of x will be 1 mod p. So they can be prime only if f(1) = f(p + 1) = f(2p + 1) = ..., which is possible only if f is the constant polynomial f(x) = p.
Thanks for that. I had said there could be a few other types of exception. But generalizing your argument:

Pick any n such that f(n)=p where p is prime. Then f(n+p), f(n+2p), etc will all be divisible by p since each power of x will be the same mod p as it is for f(n). So most values of n will generate a composite (unless it's a constant polynomial which can only generate 1 prime).

Chris

2016-03-24, 19:01   #24
ET_
Banned

"Luigi"
Aug 2002
Team Italia

3×1,597 Posts

Quote:
 Originally Posted by paulunderwood The quadratic x^2+1 has not been proven to generate an infinite number of primes. See 30 minutes into: https://www.youtube.com/watch?v=rwH-5VhBPGc and if you have the time the whole video is worth watching. Oh I see x is not an integer???
Wonderful video, thaank you

2016-03-26, 01:16   #25
MattcAnderson

"Matthew Anderson"
Dec 2010
Oregon, USA

22·3·53 Posts

Hi Math People

Attached are slides for my write-up for n^2 + n + 41. I have removed all reference to the word bifurcation. That was incorrect. The points in the graph of divisors lie on parabolas.

Regards
Matt
Attached Files
 A prime producing quadratic expression 4.pdf (428.4 KB, 145 views)

 2016-04-19, 08:35 #26 bhelmes     Mar 2016 23·37 Posts A peaceful day for all members there are some "new" results for the polynomial f(n)=n^2+n+41 for n<=2^35 http://www.devalco.de/basic_polynomi...a=1&b=1&c=41#7 For people who are not familiar with quadratic prime generators: It is possible to make a sieving construction for quadratic irreducible polynomials like f(n)=n² +1. The Sieve of Eratosthenes is a more specific variation of this construction. Normally the primes with p=f(n) or p|f(n) appear double periodically on the polynomial. If you divide the appearing primes you can be sure that there only rest a prime or the number one. Would nice to have a little feedback. The topic is quite interesting for people who look for some prime generators. An overview is in the web under http://devalco.de/#106 Have a lot of fun with the primes Bernhard
 2016-04-30, 07:47 #27 MattcAnderson     "Matthew Anderson" Dec 2010 Oregon, USA 22·3·53 Posts Hi Math People, Thank you for considering my little project. As I have state before the word "bifurcation" was incorrectly used by me to describe a graph. New words are graph of discrete divisors. For what it is worth. Also, Bernhard, your work on seiving and quadratic functions is very interesting. Regards, Matthew Attached Thumbnails
2017-04-25, 12:42   #28
MattcAnderson

"Matthew Anderson"
Dec 2010
Oregon, USA

11748 Posts
primes of the form n^2+n+41

Hi all,

I continue to doodle with h(n). I define h(n) as n^2 + n + 41. We assume n is a positive integer. So far we know that no primes less than 40 ever divide h(n). Also, there are no positive integers n that make 59 divide h(n).

We prove these two facts by exhaustive search in a residue table.

Some of my progress is shown at this webpage -

We know that if n is congruent to x mod y and (x,y) is on the curve

p(r,c) = (c*x – r*y)2 – r*(c*x – r*y) – x + 41*r^2 = 0

and 0<r<c, c>1 and gcd(r,c) = 1 and all four of r,c,x, and y are integers,

then h(n) is a composite number.

further, all n such that h(n) is a composite number are probably on p(r,c)=0.

Each such pair (r,c) yields integer points on a parabola.

Possible next steps -
We want to show that h(n) is prime an infinite number of times. p(r,c) is 0 for an infinite number of values x and y. Given that x and y are counting numbers and restrict x and y to be not a solution of p(r,c) = 0, can we infer that there are still an infinite set of pairs (x,y) such that n = x mod y will give us a prime value for h(n)?

This seems to be a hard question.

Regards,
Matt
Attached Files
 extended resedue table.pdf (118.3 KB, 107 views) extended resedue table explaination.txt (415 Bytes, 91 views) prime producing polynomial continues.pdf (148.1 KB, 96 views)

 2017-10-16, 00:47 #29 MattcAnderson     "Matthew Anderson" Dec 2010 Oregon, USA 22·3·53 Posts Hi all, There are two webpages that I have made regarding Prime Producing Polynomial. The webpages are - sites.google.com/site/mattc1anderson/prime-producing-polynomial sites.google.com/site/primeproducingpolynomial Regards, Matt
 2020-11-03, 22:58 #30 MattcAnderson     "Matthew Anderson" Dec 2010 Oregon, USA 22×3×53 Posts Hi again all, I added a counter example to a conjecture of mine that i about 2 years old. Regarding our Prime Producing Polynomial Project for the enhancement of mathematical trivia database Specifically, counterexample to conjecture about x2_plus_x_plus_41.pdf Cheers, Matt Last fiddled with by MattcAnderson on 2020-11-03 at 22:59

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