Prove infinitely many prime of the form 6k+5

In summary, the conversation discusses how to prove that there are infinitely many primes of the form 6k+5, where k is a nonnegative integer. A proof by contradiction is suggested and the use of a result about arithmetic progressions with relatively prime numbers is mentioned. A simpler proof is also suggested.
  • #1
Hwng10
4
0

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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  • #2
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
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
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.
 

What is the meaning of "infinitely many prime of the form 6k+5"?

The phrase "infinitely many prime of the form 6k+5" refers to a mathematical concept known as the "twin prime conjecture." It states that there are infinitely many pairs of prime numbers that differ by 2, and one of the numbers in each pair is of the form 6k+5.

Why is it important to prove that there are infinitely many prime numbers of the form 6k+5?

Proving the existence of infinitely many prime numbers of the form 6k+5 would provide evidence for the twin prime conjecture and contribute to our understanding of the distribution of prime numbers. It could also lead to further developments in number theory and other mathematical areas.

What is the current status of the proof for infinitely many prime numbers of the form 6k+5?

As of now, there is no formal proof for the twin prime conjecture or the existence of infinitely many prime numbers of the form 6k+5. However, many mathematicians have made progress towards proving it, and there are several potential approaches that are being explored.

What are some possible strategies for proving the existence of infinitely many prime numbers of the form 6k+5?

Some strategies that have been proposed include using advanced techniques from number theory, such as the Goldbach conjecture or the generalized Riemann hypothesis, as well as exploring the connections between primes and other mathematical concepts such as algebraic geometry and modular forms.

What are some challenges that make it difficult to prove the existence of infinitely many prime numbers of the form 6k+5?

Some challenges include the complexity of number theory and the limited understanding of prime numbers, as well as the lack of a unified approach that can be applied to all cases. Additionally, there may be other factors or patterns that are not yet known, making it difficult to determine a definitive proof.

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