Do Prime Numbers Follow a Pattern?

In summary: Mathworld:http://mathworld.wolfram.com/LuckyNumberofEuler.htmlhas an explanation in terms of Heegner numbers, but I'm not nearly smart enough to understand it :smile:Nor me!In summary, there is no known pattern or rule for calculating prime numbers, but with a little thought you should be able to easily find a counter-example without laboriously going through numbers.
  • #1
Raschedian
11
5
Hello everyone!

I was going through a simple high school level mathematics book and got to the following question:

n2 - n + 41 is a prime for all positive integers n.

You're supposed to find a counter-example and prove the statement false.

You could of course sit and enter different values for n until you get a composite number and then use that value of n as the counter-example.

But is there a way to find some pattern or rule for prime or composite numbers so that you don't have to do the work manually? This is probably a trivial question but I got curious. Thank you!
 
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  • #2
There is no known pattern or rule for calculating prime numbers, but with a little thought you should be able to easily find a counter-example without laboriously going through numbers.
 
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  • #3
phyzguy said:
There is no known pattern or rule for calculating prime numbers, but with a little thought you should be able to easily find a counter-example without laboriously going through numbers.
Thank you!
 
  • #4
Raschedian said:
I was going through a simple high school level mathematics book and got to the following question:

n2 - n + 41 is a prime for all positive integers n.

You're supposed to find a counter-example and prove the statement false.
It's pretty easy to find a counterexample for the formula above, if you think about it. A similar formula is the following: n2 + n + 41. This one also appears to generate prime numbers. It's a little harder than the first formula to spot why not all of its values are primes.
 
  • #5
Uh? Simple proof? Hmm... how come... hmmm... hmmmmm...

Ah! It's the same reason why ##n^2-n+11## also doesn't generate prime numbers! Haha, smart problem!
 
  • #6
Try writing it as n(n-1) + 41. Is there a vale of n that makes it obvious this is not prime?
 
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  • #7
I have once checked one of the polynomials up to ##n=100## and there are indeed disproportionately many primes as results. Does anyone know why? I mean in terms of Legendre symbols or so?
 
  • #8
phyzguy said:
Try writing it as n(n-1) + 41. Is there a vale of n that makes it obvious this is not prime?

The way I thought of this was rewriting as ##n^2 + (41-n)##
 
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  • #9
fbs7 said:
The way I thought of this was rewriting as ##n^2 + (41-n)##
Yes, that's probably the most straightforward way.
 
  • #10
fbs7 said:
The way I thought of this was rewriting as ##n^2 + (41-n)##
That's what I thought also, but @phyzguy's approach allowed me to see the solution to
Mark44 said:
A similar formula is the following: n2 + n + 41. This one also appears to generate prime numbers. It's a little harder than the first formula to spot why not all of its values are primes.
 
  • #11
Hmm... I'm kinda missing the ##n*(n-1)+41##... how does that make it non-prime... hmm... 21*20?... 11*10?...hmm... can't be that... 41*40?... oh.. .41*40+41! hahaha... I got it! :biggrin: ... I'm so slow

It's actually the same thing as ##n*(n+1)+41##, I guess!
 
  • #12
fbs7 said:
Hmm... I'm kinda missing the ##n*(n-1)+41##... how does that make it non-prime... hmm... 21*20?... 11*10?...hmm... can't be that... 41*40?... oh.. .41*40+41! hahaha... I got it! :biggrin: ... I'm so slow

It's actually the same thing as ##n*(n+1)+41##, I guess!
Well, maybe.
As a hint, consider that ##n^2 + n + 41 = n^2 + n + 40 + 1##, where the latter expression can be written as a perfect square trinomial for some value of n.
 
  • #13
10.

(Didn't read the other comments.)
 
  • #14
AdamF said:
10.
?
AdamF said:
(Didn't read the other comments.)
 
  • #15
Mark44 said:
?
Yes, a commentary which reminds me not to write what I think.
AdamF said:
10.

(Didn't read the other comments.)
Maybe someone should tell him that ##10^2\pm 10+41## are both prime.
 
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  • #16
I read it as "n^2 - n - 41".

-- The way it's written is more immediate, though.

Reduce the last two terms to zero by setting n equal to the quantity you're subtracting.
In this case, take:
N = 41
 
  • #17
Just let n=41.
 
  • #18
It seems worthwhile checking to see, for a more general expression if you can find a way , a value , to make the expression be even or end in 5, as the simplest cases. Here , if n is odd, the sum is odd, same for ifn is even. Five will not work. Just mentioning as a general technique. Usually going by using the remainder of a well-chosen number should help.
 
  • #19
fresh_42 said:
I have once checked one of the polynomials up to ##n=100## and there are indeed disproportionately many primes as results. Does anyone know why? I mean in terms of Legendre symbols or so?
Mathworld:
http://mathworld.wolfram.com/LuckyNumberofEuler.html
has an explanation in terms of Heegner numbers, but I'm not nearly smart enough to understand it :smile:
 
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  • #20
Nor me!
##\rm The~core~equation~for~~~##x2+x+41
##\rm~~~~~~~~~~~~~~~is~~~~~~~~~~~~~~~~##x2+163
this will be divisible by 163 at n=81
of course, at n=40 , i.e. 40 x 41 [x1]
##~~~~~~~~~~~~## and n=41##~~~~~~~## 41 x 42 ##~~ "##
the first composites ( of 41) are produced at and then below the square root of the function ( of x)
 
  • #21
Janosh89 said:
Nor me!
##\rm The~core~equation~for~~~##x2+x+41
##\rm~~~~~~~~~~~~~~~is~~~~~~~~~~~~~~~~##x2+163
What is a "core equation" and why is ##x^2 + 163## a core equation for ##x^2 + x + 41##? Also note that neither ##x^2 + x + 41## nor ##x^2 + 163## is an equation.
Janosh89 said:
this will be divisible by 163 at n=81
What will be divisible by 163? Without some elaboration, it would be difficult to see that ##81^2 + 163 = 81^2 + 2(81) + 1 = (81 + 1)^2##, but what does this have to do with ##x^2 + x + 41##?
Janosh89 said:
of course, at n=40 , i.e. 40 x 41 [x1]
##~~~~~~~~~~~~## and n=41##~~~~~~~## 41 x 42 ##~~ "##
the first composites ( of 41) are produced at and then below the square root of the function ( of x)
The square root of what?
Please try to be clearer in your replies.
 
  • #22
I will post, y=x2+163 in future
##81^2+81+41=163×41##
##y=x^2+x+41##
##∴y=163×41~ when~x=81##
 
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  • #23
alan2, back in post 17 gave the obvious answer: "let x= 41"! If x= 41, [tex]x^2+ x+ 41= (41)^2+ 41+ 41= 41(4`1+ 41+ 41)= 41(123)[/tex].
 
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  • #24
Yes , it was explicit.
 

1. Do prime numbers follow a specific pattern?

There is no known pattern for the distribution of prime numbers. Prime numbers are considered to be random and unpredictable, making it difficult to find a pattern or formula to generate them.

2. Can the distribution of prime numbers be predicted?

No, the distribution of prime numbers cannot be predicted. While there are some patterns that may appear in the distribution of primes, they are not consistent enough to be used for prediction.

3. Are there any exceptions to the pattern of prime numbers?

Yes, there are some exceptions to the pattern of prime numbers. For example, the number 2 is the only even prime number, and there are also some prime numbers that follow a specific pattern, such as Mersenne primes.

4. Is there a formula for generating prime numbers?

There is no known formula for generating prime numbers. While there are some algorithms that can generate a list of prime numbers, they are not considered to be a formula as they are based on trial and error rather than a specific pattern.

5. Why is it important to study the pattern of prime numbers?

Studying the pattern of prime numbers is important for understanding the fundamental properties of numbers and for developing encryption methods in computer science. It also helps in identifying patterns and relationships between numbers and in solving mathematical problems.

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