# Prove infinitely many prime of the form 6k+5

1. Nov 25, 2011

### Hwng10

1. The problem statement, all variables and given/known data
Prove that there are infinitely many prime of the form 6k+5, where k is nonnegative integer.

2. Relevant equations

3. The attempt at a solution
Prove by contradiction. Suppose there are finitely many prime of the form 6k+5. Then
i get stucked. Anyone can help me ??

2. Nov 26, 2011

### Bacle2

Well, there isa result that any arithmetic progression an=a0+nr
with a0 and r relatively prime contains infinitely-many primes. Is that the type of proof you want (adapted to a0=5 and r=6)?

3. Nov 26, 2011

### Dick

You got "stucked" before you really got started. Suppose M=p1*p2*...*pk where the p's are your primes. Think about the prime factorization of 3*M+2. Can you show none of the p's are factors? Can you show at least one of the factors must be equal to 5 mod 6?

4. Nov 26, 2011

### Dick

That proof is way too hard. There are simpler proofs for special cases. This is one of them.