(a) State Dirichlet's theorem on primes in arithmetic progression.

(b) Let $d$ be the discriminant of a binary quadratic form, and let $p$ be an odd prime. Show that $p$ is represented by some binary quadratic form of discriminant $d$ if and only if $x^{2} \equiv d(\bmod p)$ is soluble.

(c) Let $f(x, y)=x^{2}+15 y^{2}$ and $g(x, y)=3 x^{2}+5 y^{2}$. Show that $f$ and $g$ each represent infinitely many primes. Are there any primes represented by both $f$ and $g$ ?