# Thread: Infinitely many primes of the sequence 3n + 2

1. ## 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

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

Tonio

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

Tonio
Errr.. what!!?

4. 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.

5. 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

6. Originally Posted by djino
Errr.. what!!?
Is the problem that you do not know what "(mod 3)" means?