Math 620  Problem Set #8  Due Mon., Nov. 5, 2001

1. a) Show directly that there are infinitely many primes ≡ −1(mod 6).

b) Using quadratic reciprocity, show that there are infinitely many primes ≡ 1(mod 6). [Hint: For which p is there a \sqrt{−3} in ℤ/p?]

2. Consider the characters χ_{i} modulo 8, given by χ_{0}(n) = (\frac{1}{n}), χ_{1}(n) = (\frac{−1}{n}), χ_{2}(n) = (\frac{−2}{n}), χ_{3}(n) = (\frac{2}{n}), where (\frac{a}{n}) is the Jacobi symbol if n is odd, and is equal to 0 if n is even. Let L(s, χ_{i}) = ∑_{n=1}^{∞} \frac{χ_{i}(n)}{n^{s}} for each i.

a) Find a product expansion for each L(s, χ_{i}), analogous to the expansion for ζ(s). Then show that L(s, χ_{0}) diverges as s → 1^{+}, but that L(s, χ_{i}) converges as s → 1^{+} for i = 1, 2, 3. [Hint: Compare to ζ(s).]

b) For j = 1, 3, 5, 7, let P_{j} be the set of primes congruent to j modulo 8. Show that ∑_{p∈P_{1}} \frac{1}{p} diverges. [Hint: Show that ∏_{p∈P_{1}}(1 − p^{−s})^{−1} diverges as s → 1^{+}, by using a product formula for ∏_{i=0}^{3} L(s, χ_{i}) obtained via part (a).] Conclude that P_{1} is infinite.

c) Redo part (c) for P_{3}, P_{5}, and P_{7}. [Hint: Similar to part (b), but use products of the form ∏_{i=0}^{3} L(s, χ_{i})^{±1}, where the ± signs are chosen appropriately.]

3. Let d be a positive square-free integer.

a) If \frac{P_{n}}{Q_{n}} is a continued fraction convergent to \sqrt{d}, show that P_{n}^{2} − dQ_{n}^{2} ≤ 1 + 2\sqrt{d}. [Hint: Problem Set 5 #2(f).]

b) If x^{2} − dy^{2} = 1, where x, y ∈ ℕ, show |\frac{x}{y} − \sqrt{d}| < \frac{1}{2y^{2}} .

c) Show that there is an integer k with |k| < 1+2\sqrt{d} such that the equation x^{2}−dy^{2} = k has infinitely many solutions in integers. [Hint: Part (a).] Show that there are infinitely many elements α ∈ ℤ[\sqrt{d}] such that N(α) = k.

d) Show that there are finitely many residue classes of ℤ[\sqrt{d}] modulo k, and that some residue class C contains two elements α, α′ as in part (c), such that α′ ≠ ±α.

e) Show that αᾱ′ ≡ 0 (mod k) in ℤ[\sqrt{d}], where the bar indicates the image under the conjugacy map a + b\sqrt{d} ↦ a − b\sqrt{d}.

f) Let β = αᾱ′/k ∈ ℚ[\sqrt{d}]. Show that β ∈ ℤ[\sqrt{d}], that β ≠ ±1, and that N(β) = 1.

g) Conclude that there are infinitely many elements of norm 1 in ℤ[\sqrt{d}], and there are infinitely many solutions in ℤ to the Diophantine equation x^{2} − dy^{2} = 1.

4. Let ℍ be the (non-commutative) ring of quaternions α = a+bi+cj+dk, with a, b, c, d ∈ ℝ, and with the usual rules for addition and multiplication. Define ᾱ = a − bi − cj − dk and N(α) = αᾱ.

a) Write N(α) in terms of a, b, c, d, and then show that N(αβ) = N(α)N(β). (Beware of non-commutativity.)

b) Deduce that if m, n ∈ ℤ can each be expressed as the sum of four squares, then so can their product.

c) Try to express various positive integers as the sum of four squares. Conjectures? Proofs?

5. Call a subset S of the plane symmetric (about the origin) if v ∈ S ⇒ −v ∈ S. Call S convex if the line segment connecting any two points of S also lies in S. For which A > 0 can you find a symmetric convex closed set S of area A that contains no lattice points other than the origin? (You needn’t prove all other A’s fail.)