# Infinitely many primes of the sequence 3n + 2

• Nov 9th 2009, 11:47 AM
djino
Infinitely many primes of the sequence 3n + 2
I'm having trouble with showing that there are infinitely many primes in the sequence 3n + 2 (and I'd also need to follow by answering to what other sequences can this argument be applied?)

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I assume I start off by let p1 < p2 < ... < pk be a finite list of primes of a the sequence 3n +2. Then also Let N = 3p1*p2*....*pk + 2.

Not sure where to go from here?

djino
• Nov 9th 2009, 08:17 PM
tonio
Quote:

Originally Posted by djino
I'm having trouble with showing that there are infinitely many primes in the sequence 3n + 2 (and I'd also need to follow by answering to what other sequences can this argument be applied?)

---

I assume I start off by let p1 < p2 < ... < pk be a finite list of primes of a the sequence 3n +2. Then also Let N = 3p1*p2*....*pk + 2.

Not sure where to go from here?

djino

The number $\displaystyle N\equiv 2\!\!\!\pmod 3$ and since the product of primes $\displaystyle p\equiv 1\!\!\!\pmod 3$ is again of the same form, then...

Tonio
• Nov 10th 2009, 05:43 AM
djino
Quote:

Originally Posted by tonio
The number $\displaystyle N\equiv 2\!\!\!\pmod 3$ and since the product of primes $\displaystyle p\equiv 1\!\!\!\pmod 3$ is again of the same form, then...

Tonio

Errr.. what!!? (Nerd)
• Nov 10th 2009, 07:32 PM
keityo
An easy way to do this would be to use Dirichlet's theorem on arithmetic progressions. It pretty much says that if greatest common divisor of a and n is 1, then there are infinitely many primes of the form a+nd.

In your case, 2 and 3 have a greatest common divisor of 1, so the theorem applies.
• Nov 11th 2009, 01:52 AM
tonio
Quote:

Originally Posted by keityo
An easy way to do this would be to use Dirichlet's theorem on arithmetic progressions. It pretty much says that if greatest common divisor of a and n is 1, then there are infinitely many primes of the form a+nd.

In your case, 2 and 3 have a greatest common divisor of 1, so the theorem applies.

I don't think this is "the easy way" but rather the overkill way: it's clear the OP hasn't studied/can't use D.T., otherwise the question is trivial.
DT is an important theoretical tool, but proving directly that there are infinite primes of certain form can be really enlightening, and this is probably what the OP's question is meant for.
The hint is given, and the OP answering "err...what" won't really make a lot for him/her to solve the problem. Instead he/she must think a while.

Tonio
• Nov 11th 2009, 02:32 AM
HallsofIvy
Quote:

Originally Posted by djino
Errr.. what!!? (Nerd)

Is the problem that you do not know what "(mod 3)" means?