Prove infinitely many prime of the form 6k+5

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Homework Statement


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


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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 ??
 

Answers and Replies

  • #2
Bacle2
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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
Dick
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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
Dick
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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)?
That proof is way too hard. There are simpler proofs for special cases. This is one of them.
 

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