Question

Find all primes p for which −5 is a quadratic residue.

Answer 1

For some integer p, pick any integer x and compute $a=x^2(modp)$ .

$-p+a$ is a negative number that is a quadratic residue (mod p).

In fact, $pk+a$ is a quadratic residue that is a negative number as long as k is negative.

Now, we need to find $p$ such that ;

1. p is prime number
2. -5 is a quadratic residue

So, if we consider $-5$ ; then $kn-5$ will also be quadratic residue; where k is the chosen smallest integer such that $kn-5>0$ .

Now let us choose $p=7$ ; obviously it is a prime number.

Also $Z_p^*= [1,2,3,4,5,6]$

And, $4^2=16mod 7=2=(-1)*7+2=-5$

Clearly; -5 is a quadratic residue for $p=7$

Now, for any other prime p ; -5 won't be a quadratic residue because;

$kp-5$ would never be a perfect square for any $k\epsilon Z^+$

Thus, 7 is the only prime number for which -5 is a quadratic residue.