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Old 2021-06-15, 09:08   #1
MattcAnderson
 
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"Matthew Anderson"
Dec 2010
Oregon, USA

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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.
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Old 2021-06-16, 07:42   #2
MattcAnderson
 
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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
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File Type: pdf small equation coefficient doublecheck 33.pdf (631.8 KB, 42 views)

Last fiddled with by MattcAnderson on 2021-06-16 at 13:35 Reason: explained method
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Old 2021-08-15, 21:38   #3
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"Matthew Anderson"
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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
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File Type: pdf Prime Producing Polynomial August 2021.pdf (286.6 KB, 28 views)
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Old 2021-09-03, 13:47   #4
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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
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File Type: pdf count of prime values of n^2+n+41.pdf (116.5 KB, 22 views)
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Old 2021-09-03, 13:51   #5
MattcAnderson
 
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"Matthew Anderson"
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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
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Old 2021-09-03, 16:14   #6
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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.
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Old 2021-09-04, 14:22   #7
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What is the natural density of A056561? Is it zero? Or positive?
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Old 2021-09-23, 14:05   #8
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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 non-negative 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
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Old 2021-09-23, 15:46   #9
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Quote:
Originally Posted by MattcAnderson View Post
My guess is that the natural density is greater than zero.
according to the Bateman-Horn conjecture, the number of n less than or equal to X for which n2 + n + 41 is prime is asymptotically c*Li(X) for a positive constant c; this is asymptotically c*X/ln(X). This indicates a natural density of 0, though the decrease in density up to X decreases very slowly with X.

If N > 163 the density of n for which n2 + n + 41 is not divisible by any prime < N is

\frac{162}{163}\prod_{p<N,\(\frac{p}{163}\)=+1}\(1 - \frac{2}{p}\).

where \(\frac{p}{163}\) 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 2021-09-23 at 18:58 Reason: fignix topsy
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