We can use Theorem 11.2.3 to prove that there are infinitely many primes of the form 4k + 1 (where k ∈ Z). Begin by assuming that there are only finitely many such primes, and call them p1, p2, …, pm. Let N = 4 · (p1 · p2 · … · pm)² + 1, and let q be a prime number such that q | N. a. Prove that q ≡ 1 (mod 4). [Hint: Use Theorem 11.2.3.] b. Show that q ≠ pj for every j = 1, 2, …, m. c. Why does it follow from parts a and b that there are infinitely many primes of the form 4k + 1?