a) Let p be a prime number with p>3. Prove that sum of quadratic residues modulo p is divisible by p.
b) Let p be a prime number with p>5. Prove that the sum of the squares of the quadratic nonresidues modulo p is divisible by p.
a) Let p be a prime number with p>3. Prove that sum of quadratic residues modulo p is divisible by p.
b) Let p be a prime number with p>5. Prove that the sum of the squares of the quadratic nonresidues modulo p is divisible by p.
Let $\displaystyle a$ be a primitive root.
Then all quadradic residues are given by $\displaystyle 1,a^2,a^4,...,a^{p-2}$ i.e. even exponents.
The sum is $\displaystyle 1+a^2+a^4+...+a^{p-2} = \frac{1-a^p}{1-a^2} \equiv 0$ (see how we use assumption $\displaystyle p>3$).
In the second problem there is not need to have $\displaystyle p>5$. Let $\displaystyle A$ be the residue sum and $\displaystyle B$ be the non-residue sum. Then we know $\displaystyle A + B = 0$. Since $\displaystyle p$ divides $\displaystyle A$ it must means that $\displaystyle p$ divides $\displaystyle B$.