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Math Help - Legendre Symbol

  1. #1
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    Legendre Symbol

    Evaluate the Legendre symbol below:

    (461/773)

    Show all work/theorems used to compute it.
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  2. #2
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    Quote Originally Posted by math19091 View Post
    Evaluate the Legendre symbol below:

    (461/773)

    Show all work/theorems used to compute it.
    Note 461 = 1 (mod 4).

    Quadradic Reciprocity:
    (773/461)

    = (312/461) = (8/461)*(3/461)*(13/461)
    =(2/461)*(3/461)*(13/461)

    Theorem 2 (2/p)=1 iff p=1,7(mod 8) for odd primes p.

    Use theorem,
    (-1)*(3/461)*(13/461)

    Theorem 1 (3/p)=1 iff p=1,11(mod 12) for odd primes p.

    Use theorem,
    (-1)(-1)(13/461)=(13/461)

    Note that p=1(mod 4).
    Quadradic Reciprocity:
    (461/13) = (6/13) = (2/13)*(3/13)

    Use Theorem 1 and Theorem 2:
    (-1)(+1)=-1.

    Thus the Legendre Symbol is equal to -1.
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  3. #3
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    Quote Originally Posted by ThePerfectHacker View Post
    Note 461 = 1 (mod 4).

    Quadradic Reciprocity:
    (773/461)

    = (312/461) = (8/461)*(3/461)*(13/461)
    =(2/461)*(3/461)*(13/461)

    Theorem 2 (2/p)=1 iff p=1,7(mod 8) for odd primes p.

    Use theorem,
    (-1)*(3/461)*(13/461)

    Theorem 1 (3/p)=1 iff p=1,11(mod 12) for odd primes p.

    Use theorem,
    (-1)(-1)(13/461)=(13/461)

    Note that p=1(mod 4).
    Quadradic Reciprocity:
    (461/13) = (6/13) = (2/13)*(3/13)

    Use Theorem 1 and Theorem 2:
    (-1)(+1)=-1.

    Thus the Legendre Symbol is equal to -1.
    Thanks TPH. Question: how did you do this part

    (773/461)

    = (312/461) = (8/461)*(3/461)*(13/461)

    I don't see how you go from (773/461) to (312/461)
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    Quote Originally Posted by math19091 View Post
    Thanks TPH. Question: how did you do this part

    (773/461)

    = (312/461) = (8/461)*(3/461)*(13/461)

    I don't see how you go from (773/461) to (312/461)
    i believe he made the observation that 773 is congruent to 312 mod 461, so he can use either one to find the symbol, working with smaller numbers is always better

    i think one of his theorems said he could do such a manuevor
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  5. #5
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    Quote Originally Posted by math19091 View Post
    Thanks TPH. Question: how did you do this part

    (773/461)

    = (312/461) = (8/461)*(3/461)*(13/461)

    I don't see how you go from (773/461) to (312/461)
    I was assuming you know the fundamental properties of the Legendre symbol.

    Theorem: Let p be an odd prime and a and b be two integers such that gcd(a,p)=gcd(b,p)=1. If a = b (mod p) then (a/p)=(b/p).

    Thus, I just reduced the Legendre symbol by modular arithmetic.
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