Prove infinitely many prime of the form 6k+5

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Homework Help Overview

The problem involves proving that there are infinitely many prime numbers of the form 6k+5, where k is a nonnegative integer. This falls under the subject area of number theory, specifically concerning the distribution of prime numbers.

Discussion Character

  • Exploratory, Assumption checking, Mathematical reasoning

Approaches and Questions Raised

  • The original poster attempts a proof by contradiction but expresses difficulty in progressing. Some participants suggest considering the implications of the prime factorization of a specific expression related to the primes in question.

Discussion Status

Participants are exploring different approaches to the proof, including references to known results about arithmetic progressions containing infinitely many primes. There is no explicit consensus on a single method, but various lines of reasoning are being discussed.

Contextual Notes

There is mention of the complexity of certain proofs and the possibility of simpler approaches for special cases, indicating that the original poster may be seeking a more accessible method.

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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 ??
 
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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)?
 
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?
 
Bacle2 said:
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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