Additional Exercises

1. Suppose that p and q are relatively prime positive integers. Show that if cos pα, cos qα, and cos(p + q)α are all rational, then cos α is rational.

*2. Suppose that p and q are relatively prime positive integers. Show that if cos pα and cos qα are rational, then cos α is rational or α is a multiple of π/6.

3. Suppose we extend the definition of divisibility as follows: Let m be a non-zero integer, and let a/b be a rational number in lowest terms, i.e., (a, b) = 1. We say that m divides a/b, and write m|a/b, if (m, b) = 1 and m|a. Suppose also that for rational numbers r, s with denominators relatively prime to m, we say that r ≡ s (mod m) if m|(r − s).

(a) Let r, R, s, S be rational numbers whose denominators are relatively prime to m. Show that if r ≡ R (mod m) and s ≡ S (mod m), then r + s ≡ R + S (mod m) and rs ≡ RS (mod m).

(b) Suppose that r = a/b is a rational number with (b, m) = 1. Choose an integer \overline{b} such that b\overline{b} ≡ 1 (mod m). Show that r ≡ a\overline{b} (mod m). Show that if c is an integer such that r ≡ c (mod m), then c ≡ a\overline{b} (mod m).

(c) Suppose that a, A, b, B are integers such that a ≡ A (mod m), b ≡ B (mod m), and (b, m) = 1. Show that a/b ≡ A/B (mod m).

4. Let f(z) = ∑_{i=0}^{∞} a_{i}z^{i} and g(z) = ∑_{i=0}^{∞} b_{i}z^{i} be two formal power series with integral coefficients. (By “formal” we mean that we don’t care whether the radius of convergence is positive.) Define f ≡ g (mod m) if a_{i} ≡ b_{i} (mod m) for all i = 0, 1, . . ..

(a) Suppose that f, F, g, G are power series with integral coefficients. Show that if f ≡ F (mod m) and g ≡ G (mod m), then f + g ≡ F + G (mod m) and fg ≡ F G (mod m).

(b) Suppose that f and F are power series with integral coefficients, both of which have constant term 0. Show that if f ≡ F (mod m), then

\frac{1}{1 + f} ≡ \frac{1}{1 + F} (mod m).

5. Let A be an n × n matrix with integral elements. Show that if (det(A), m) = 1, then for any (c_{1}, c_{2}, …, c_{n}) ∈ Z^{n} there is a unique n-tuple (x_{1}, x_{2}, …, x_{n}) of residue classes (mod m) such that

(∗)  ∑_{j=1}^{n} a_{ij}x_{j} ≡ c_{j} (mod m)

for i = 1, 2, …, n. Also, show conversely that if (det(A), m) > 1, then the number of solutions of (∗) is either 0 or > 1, depending on the choice of (c_{1}, c_{2}, …, c_{n}).

Additional exercises

6. We recall that the binomial coefficient ({x \atop k}) is a polynomail in x, namely ({x \atop k}) = x(x − 1) ⋯ (x − k + 1)/k!. Thus ({−1/3 \atop k}) is a rational number, say

({−1/3 \atop k}) = \frac{a_{k}}{q_{k}}

where (a_{k}, q_{k}) = 1 and q_{k} > 0.

(a) Show that q_{k} is a power of 3.

(b) Show that a_{k} is odd if and only if k can be written as a sum of distinct powers of 4.

7. (Nicol–Selfridge) Show that if n is an odd integer > 3, then there is a prime number p such that p ∤ n and p|(2^{φ(n)} − 1).