Originally Posted by
ThePerfectHacker 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.