# Quadratic Residues, stating at least $2,5$ or $10$.

If $p>5$ is a prime, then show that at least one of $2,5$ or $10$ is a quadratic residue. Then suppose, $p$ is an odd prime and $b$ is the smallest positive quadratic non residue mod $p$. prove that $b$ is also prime.

I went about it this way, but you can see where my assumptions lie pretty weak in saying that $a,b$ is equal to $2$ and $5$. Here it is...

Let $a$ and $b$ be integers not divisible by prime $p>5$. such that. $$\left(\frac{a}{p}\right)\left(\frac{b}{p}\right)=\left(\frac{ab}{p}\right)$$ Therefore if $\left(\frac{ab}{p}\right) = 1$ then $\left(\frac{a}{p}\right)=\left(\frac{b}{p}\right)$ so either both are residues mod $p$ or both are non residues.

If $\left(\frac{ab}{p}\right) = 1$, then $\left(\frac{a}{p}\right)=-\left(\frac{b}{p}\right)$, so exactly one of $(a/p)$ and $(b/p)$ is a quadratic reside. FIN

I feel like i'm onto something here, don't' really want to give up on it. I need to somehow translate, $a=2$ and $b=5$, and $ab=10$. Is there a direct way to show this. From this its obvious that $b$ is a prime number, but also having difficulty showing that due to my first difficulty. Thanks!!

• $2 \cdot 5 = 10.$ Dec 19, 2013 at 5:17
• I don't understand your confusion. Just let $a=2, b=5$. Dec 19, 2013 at 5:22
• without symbols, if $2$ and $5$ are non-residues, then $(2)(5)$ is a quadratic residue. Dec 19, 2013 at 5:26

Hint: Let $q$ be the smallest positive integer which is a non-residue. If $q$ is not prime, then $q=ab$ for positive integers $a,b\gt 1$. These are both residues, by the definition of $k$. But then $\dots$.
Suppose none of $$2$$,$$5$$, or $$10$$ are quadratic residues. Then the Legendre symbol of each of these mod $$p$$ is $$-1$$. However, $$10=2\cdot5$$ so the Legendre symbol of $$10$$ is $$(-1)\cdot(-1)=(+1)$$ contradicting that none are quadratic residues, i.e. at least one is.