3. (i) Find all prime numbers for which -3 is a quadratic residue;
(ii) Find all prime numbers for which ±3 is a quadratic residue;
(iii) Find all prime numbers for which ±3 is a quadratic non-residue;
(iv) Find all prime numbers for which 3 is a quadratic residue and -3 is a quadratic non-residue;
(v) Find all prime numbers for which 3 is a quadratic non-residue and -3 is a quadratic residue;
(vi) Find the prime factorization of $(100)^{2}-3$ and $(150)^{2}+3$.
3. (i) $p \equiv 1(\bmod 6)$. (ii) $p \equiv 1(\bmod 12)$. (iii) $p \equiv 5(\bmod 12)$; (iv) $p \equiv-1$ $(\bmod 12) ;(\mathrm{v}) p \equiv-5(\bmod 12) .(\mathrm{vi})(100)^{2}-3$ 的素因数 $p \equiv \pm 1(\bmod 12)$. $100^{2}-3=13 \cdot 769 ; 150^{2}+3$ 的素因数 $p \equiv 1(\bmod 6)$ 及 $p=3$.
$$150^{2}+3=3 \cdot 13 \cdot 577$$

3. (i) $p \equiv 1(\bmod 6)$. (ii) $p \equiv 1(\bmod 12)$. (iii) $p \equiv 5(\bmod 12)$; (iv) $p \equiv-1$ $(\bmod 12) ;(\mathrm{v}) p \equiv-5(\bmod 12) .(\mathrm{vi})$ The prime factors of $(100)^{2}-3$ are $p \equiv \pm 1(\bmod 12)$. $100^{2}-3=13 \cdot 769 ;$ The prime factors of $150^{2}+3$ are $p \equiv 1(\bmod 6)$ and $p=3$.
$$150^{2}+3=3 \cdot 13 \cdot 577$$