Prove infinitely many prime of the form 6k+5

[Hwng10]

Homework Statement

Prove that there are infinitely many prime of the form 6k+5, where k is nonnegative integer.

Homework Equations

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 ??

 

[Bacle2]

Well, there isa result that any arithmetic progression a_n=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)?

 

[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?

 

[Dick]

Well, there isa result that any arithmetic progression a_n=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)?

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