Evaluate the Legendre symbol below:

(461/773)

Show all work/theorems used to compute it.

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- May 5th 2007, 01:07 PMmath19091Legendre Symbol
Evaluate the Legendre symbol below:

(461/773)

Show all work/theorems used to compute it. - May 5th 2007, 06:27 PMThePerfectHacker
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. - May 10th 2007, 09:44 AMmath19091
- May 10th 2007, 09:52 AMJhevon
- May 10th 2007, 10:15 AMThePerfectHacker
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.