# Thread: Showing a prime in an arithmetic progression

1. ## Showing a prime in an arithmetic progression

Hello I am stuck on this problem and I need some help on how to do this.

Show that if p is a prime in the arithmetic progression 3n+1, n = 1,2,3,.... then it is also in the arithmetic progression 6n+1, n=1,2,3,.......

2. ## Re: Showing a prime in an arithmetic progression

Start with an assumption. Assume that $3n+1$ is prime. What do you know about all primes that are bigger than $3(1)+1$?

3. ## Re: Showing a prime in an arithmetic progression

First, any number in the sequence 6n+ 1= 3(2n)+ 1 is also in the sequence 3n+1. To go the other way, from 3n+ 1 to 6m+ 1, we must have n= 2m, that is, n is even. 3n+ 1, with n odd, is even so every prime in the series 3n+1, which must be odd is of that form.

4. ## Re: Showing a prime in an arithmetic progression

Originally Posted by HallsofIvy
First, any number in the sequence 6n+ 1= 3(2n)+ 1 is also in the sequence 3n+1. To go the other way, from 3n+ 1 to 6m+ 1, we must have n= 2m, that is, n is even. 3n+ 1, with n odd, is even so every prime in the series 3n+1, which must be odd is of that form.
And that's what I was getting at with my question about primes bigger than 3(1)+1. They are all odd.

5. ## Re: Showing a prime in an arithmetic progression

Oh I see now thank you so much guys!