# Showing a prime in an arithmetic progression

• Oct 8th 2013, 01:53 PM
gfbrd
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,.......
• Oct 8th 2013, 01:59 PM
SlipEternal
Re: Showing a prime in an arithmetic progression
Start with an assumption. Assume that \$\displaystyle 3n+1\$ is prime. What do you know about all primes that are bigger than \$\displaystyle 3(1)+1\$?
• Oct 8th 2013, 02:04 PM
HallsofIvy
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.
• Oct 8th 2013, 02:23 PM
SlipEternal
Re: Showing a prime in an arithmetic progression
Quote:

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.
• Oct 8th 2013, 02:34 PM
gfbrd
Re: Showing a prime in an arithmetic progression
Oh I see now thank you so much guys!