Determining prime numbers p which satisfy quadratic residues modulo p

Question

I'm learning about quadratic reciprocity and I'm stuck on an exercise. It states :

Determine the congruence characterizing all prime numbers p for the following integers such that they are quadratic residues mod p

(a) 5

(b) -5

(c) 7

(d) -7

An example they gave involving a Legendre symbol was : $(\frac{-2}{p}) = 1$ iff $p \equiv 1,3 \ (mod\ 8)$

I'm not really sure how to approach this. Is there a general method I should be following?

Do I examine the problem in terms of Legendre symbols and their properties?

Answer 1

You may use quadratic reciprocity law. I'll show the general method here by working $(a)$

Since $5\equiv 1\pmod{4}$ we have $$\left(\dfrac{5}{p}\right) = \left(\dfrac{p}{5}\right)$$ The quadratic residues of $5$ are $0,1,4$ and nonresidues are $2,3$. So $$\left(\frac{5}{p}\right) = \left\{ \begin{array}{} 1 &:&p\equiv 1,4 \pmod{5}\\-1&:&p\equiv 2,3\pmod{5}\end{array}\right.$$