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Math Help - Question #2: Prime

  1. #1
    Mel
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    Question Question #2: Prime

    Adapt the proof of Theorem 1.5.9 to prove:

    Theorem 1.5.9 - There is an infinit number of primes.

    (a) There is an infinite number of primes of the form 4n + 3.

    (b) There is an infinite number of primes of the form 6n + 5.
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  2. #2
    MHF Contributor chiph588@'s Avatar
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    Dirichlet's theorem on arithmetic progressions

    Use Dirichlet's theorem on arithmetic progressions.
    This website should tell you all you need to know

    Dirichlet's theorem on arithmetic progressions - Wikipedia, the free encyclopedia
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    Thanks, that helped alot. Another question. Is there another way to show a number is prime other than just being divisible by another number.
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    MHF Contributor chiph588@'s Avatar
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    yes there is, using modular arithmetic

    Wilson's Theorem states that (p-1)! = -1 mod p if and only if p i prime

    also using fermat's little theorem if a^(n-1) = 1 mod n, then n is said to have a strong probability of being prime.
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    Can I have an example using the number 301.
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  6. #6
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    Using Dirichlet's theorem is complete overkill here.

    Quote Originally Posted by Mel View Post
    (a) There is an infinite number of primes of the form 4n + 3.
    Assume there are fininitely many prime of that form p_1,...,p_n.
    Let N = 4(p_1\cdot .... \cdot p_n) - 1.
    This odd number can be factored into prime divisors.
    It cannot be that each prime divisor has form 4k+1 because the product of numbers of the form 4k+1 still has the form 4k+1. Therefore, there must be a prime divisor of the form 4k+3. Thus, N be divisible by one of p_1,...,p_n. But that implies p_i | 1 for 1\leq i \leq n. And this is impossible.

    (b) There is an infinite number of primes of the form 6n + 5.
    I think this is similar to the one above. Try modifing the proof.
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