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Math Help - Dirichelet's Theorem

  1. #1
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    Dirichelet's Theorem

    I was working on Dirichelet's Theorem, but I can't quite get it. Perhaps I am missing something.

    Dirichelet's Theorem says that for positive integers a and b, if gcd(a, b) = 1 then the airthmetic progression: a, a + b, a + 2b, a + 3b, ... will have an infinite number of primes.

    I can come up with a proof using a special case; that is:

    Let a = 3, and b = 4
    The arithmetic progression then is 3, 3 + 4, 3 + 2*4, 3 + 3*4, ...

    Proof (by contradiction)

    The above implies that there exists an infinite number of primes of the form 4n + 3, say p_0 = 3, p_1, p_2, ..., p_r in increasing order.

    Consider Q = 4*p_1*p_2*...*p_4 + 3
    Certainly, Q > p_1 and Q = 4*integer + 3 says that Q is composite. Therefore, some prime < Q must divide Q.
    Q is odd, and thus prime divisors of Q must be modd. An odd prime is of the form 4n + 1 or 4n + 3

    Claim: There exists a prime of the form 4n + 3 that divides Q

    **Proof (by contradiction):
    Assume all primes dividing Q are of the form 4*integer + 1
    Look at (4m + 1)*(4s + 1) = 4(4ms + s + m) + 1
    (4m + 1)*(4s + 1) = 16ms + 4s + 4m + 1
    Q has to be 4 int + 1
    Contradiction.

    Going back to the other proof;

    Q has to be a prime divisor, say q, of the form 4*int + 3
    So q must come from the list 3 = p_0 = p_1, ..., p_r

    Case 1: q = 3
    3|Q and 3|3 so 3|(Q - 3)

    And (Q - 3) = 4*p_1*p_2*...*p_r

    Contradiction.

    Case 2: q = p_i for some i w/ 1 <= i <= r

    q|Q and q|4*p_1*p_2*...*p_r so q|(Q - 4*p_1*p_2*...*p_r)

    (Q - 4*p_1*p_2*...*p_r) = 3

    Contradiction.

    Thus, the proof is complete.

    Obviously this is much easier than the general theorem proof, which is what I have been trying to work on.

    Thanks.
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  2. #2
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    Quote Originally Posted by AfterShock View Post
    Dirichelet's Theorem says that for positive integers a and b, if gcd(a, b) = 1 then the airthmetic progression: a, a + b, a + 2b, a + 3b, ... will have an infinite number of primes.
    Exactly. That is it.

    Here is some nice facts and history.

    The Dirchelet theorem is extremely complicated to prove. The proof is beyond almost any number theory book you pick up. Even, Hardy and Wright, they prove the Prime Number Theorem, but they state that the proof is so advanced that even their book has to skip it.

    This theorem does has some importances in number theory. The Famous Quadradic Reciprocity law, was first incompletely proven by Legendre, but he relied on the fact that infinitely many prime numbers exist in arithmetic progressions satisfing the Dirichlet criterion.

    Dirichlet is one of my favorite mathemations, it can be argued, that he was the first to intoduce analysis to number theory. The idea that Dirichlet had is to define a function called "Dirichlet L-function", (similar to the zeta function) and through it he arrived at his result. I believe this proof is from functional analysis (I am sure you heard of it). A strange approach but he managed to do it.

    Yes, specific cases, like you mentioned, can be proven rather simply. But the general case is completely different.
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