# Proving statement with Gauss's reciprocity law

State Gauss' reciprocity law for two distinct odd primes $$p$$ and $$q$$. Hence, given an odd prime $$p$$, prove carefully that the following statements are equivalent:

• $$p\neq5$$ and there is an intenger $$k$$ such that $$p$$ divides $$5-k^2$$;
• $$p$$ is congruent to $$\pm1\mod{5}$$.

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I am asked to solve the above question. I have got the Gauss's reciprocity law below.

For distinct odd primes $$p$$ and $$q$$, the theorem states: $$\operatorname{Leg}(q,p)=\begin{cases}\operatorname{Leg}(p,q)&\text{if > p\equiv1 or q\equiv1\mod4},\\-\operatorname{Leg}(p,q)&\text{if > p\equiv-1 and q\equiv-1\mod4}.\end{cases}$$

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I am stuck when I try to do the second bit. Can anyone give me some hints on how to start with the first statement?

The first statement is telling you that there is an integer $$k$$ such that $$p|5-k^2$$, this is equivalent to say that there is a $$k$$ such that $$k^2\equiv 5\mod p$$, i.e. is equivalent to say that $$5$$ is a quadratic residue mod $$p$$, i.e. equivalent to $$Leg(5,p)=1$$.
So, what you are trying to prove is $$Leg(5,p)=1\iff p\equiv \pm 1\mod 5$$. You should prove this using quadratic reciprocity. Do you see how?