Use the Law of Quadratic Reciprocity to determine if 5 is a

Use the Law of Quadratic Reciprocity to determine if 5 is a quadratic residue mod 41.

Solution

(5/41) .(41/5) = (-1)^{2 * 20) = 1

=> (5/41) = (41/5)

so 5 is quad residue mod 41 if and only if 41 is quad residue mod 5

which is indeed true 41=1 mod 5

1 is always quad residue

so 5 is quad residue mod 41

in fact 13^2 = 169 = 5 mod 41