91Ó°ÊÓ

Evaluate the following Legendre symbols: (a) \((71 / 73)\) (b) \((-219 / 383)\) (c) \((461 / 773)\) (d) \((1234 / 4567)\) (c) \((3658 / 12703)\) \([\) Hint: \(3658=2 \cdot 31 \cdot 59 .]\)

Determine whether the following quadratic congruences are solvable: (a) \(x^{2}=219(\bmod 419)\). (b) \(3 x^{2}+6 x+5 \equiv 0(\bmod 89)\). (c) \(2 x^{2}+5 x-9=0\) (mod 101).

Solve the congruence \(x^{2} \equiv 31\left(\bmod 11^{4}\right)\).

Show that 3 is a quadratic residue of 23 , but a nonresidue of 31 .