Prove infinitude of primes that satisfy these properties

[RossH]

Homework Statement

Hi. I need to prove (or at least make a conjecture) about the infinitude of primes in these cases.

1) N^2+2

2) N^2-2

3) N^2+2N+2

4) N^2+3N+2

Homework Equations

none?

The Attempt at a Solution

Already solved for number 4. This is always even, therefore there is not an infinite number of primes.

Otherwise, I don't know how I can even make a conjecture for the other ones, much less prove. The N^2+1 conjecture has not been proven yet, so I'm not sure what information I can use to solve this. Any help or hints would be much appreciated

 

[micromass]

Can you apply Fermat's little theorem on some of these questions?

 

[RossH]

Can you apply Fermat's little theorem on some of these questions?

I don't think so. FLT requires you to take the answer (mod p), but p (in this case 3 because I am squaring N) is only one possible factor. Am I missing something?

 

[micromass]

Well, you need to know when to apply it. If you want to prove that some of your numbers have an infinity of primes, then Fermat's little theorem is useless.

But maybe, you can show that some of your numbers are only prime in a finite number of cases. For example, you could show that 3 always divides the number in question. This could be done by Fermat...

 

[RossH]

Well, you need to know when to apply it. If you want to prove that some of your numbers have an infinity of primes, then Fermat's little theorem is useless.

But maybe, you can show that some of your numbers are only prime in a finite number of cases. For example, you could show that 3 always divides the number in question. This could be done by Fermat...

I don't see it. Which one are you referring to?

 

[Dick]

I don't see it. Which one are you referring to?

Try applying the divisibility by 3 approach to N^2+2. What are the possible values of N^2 mod 3?

 

[RossH]

Try applying the divisibility by 3 approach to N^2+2. What are the possible values of N^2 mod 3?

If N is even then N^2+2 is not prime, so N has to be odd. Based on the pattern it seems like an odd square can only have values of 0 or 1 (mod 3) but I'm not sure about that.

I'll try to prove it:

If m is a perfect square then m mod(3) must be 0 or 1.

If n (mod 3) =0 then n=3k for some k. Then m=9k^2 and m (mod 3) =0
If n (mod 3)=1 then n=3k+1 for some k. Then m= 9k^2+6k+1 and m (mod 3) =1
If n (mod 3)=2 then n=3k+2 for some k and m=9k^2+12k+4 congruent to 4 (mod 3)=1

So I guess my assumption was correct. Clearly the N^2 that are congruent to 1 (mod 3) are not prime in N^2+2, but what about the ones that are divisible by 3? How do I approach these when I add 2? I can think of a few off the top of my head that become prime: 9+2=11, 27+2=29. Thank you!

 

[Dick]

If N is even then N^2+2 is not prime, so N has to be odd. Based on the pattern it seems like an odd square can only have values of 0 or 1 (mod 3) but I'm not sure about that.

I'll try to prove it:

If m is a perfect square then m mod(3) must be 0 or 1.

If n (mod 3) =0 then n=3k for some k. Then m=9k^2 and m (mod 3) =0
If n (mod 3)=1 then n=3k+1 for some k. Then m= 9k^2+6k+1 and m (mod 3) =1
If n (mod 3)=2 then n=3k+2 for some k and m=9k^2+12k+4 congruent to 4 (mod 3)=1

So I guess my assumption was correct. Clearly the N^2 that are congruent to 1 (mod 3) are not prime in N^2+2, but what about the ones that are divisible by 3? How do I approach these when I add 2? I can think of a few off the top of my head that become prime: 9+2=11, 27+2=29. Thank you!

Well done. For some strange reason I was thinking N should be prime as well, hence not divisible by 3. Gah! Sorry. Back to the drawing board. Or should we just conjecture there are an infinite number? (3^3)^2+2 isn't prime. But (3^4)^2+2 is prime. So is (3^5)^2+2.

 

[RossH]

Well done. For some strange reason I was thinking N should be prime as well, hence not divisible by 3. Gah! Sorry.

Thank you. No problem. Is there any other way that you might suggest approaching this one or any of the other ones?

 

[Dick]

Thank you. No problem. Is there any other way that you might suggest approaching this one or any of the other ones?

Don't know. I do know some arithmetic sequences can be proven to contain an infinite number of primes. That's Dirichlet's theorem. I don't think the proof is at all elementary. I didn't really think all that hard about the other ones yet.

 

[atomthick]

N² + 2N + 2 = (N+1)² + 1 = M² + 1 with M a natural number

Therefore we should try to say something about:

1. N² + 2
2. N² - 2
3. N² + 1

 

[RossH]

N² + 2N + 2 = (N+1)² + 1 = M² + 1 with M a natural number

Therefore we should try to say something about:

1. N² + 2
2. N² - 2
3. N² + 1

Isn't N^2+1 unprovable?

 

[atomthick]

There is a conjecture due to Hardy saying there are infinitely many prime numbers of the forms n²+m² and n²+m²+1. But it's not proved.