Contents

1 Infinite Continued Fractions 3 2 Binary quadratic forms 5 3 Cyclotomic units 7

3.1 Finite sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 4 Study plans 9 Index 9

CONTENTS  CONTENTS

2

Chapter 1

Infinite Continued Fractions

Continued fractions have many applications, from the abstract to the concrete. They give good rational approximations to irrational numbers, and the have been used to understand why you can’t tune a piano perfectly.^{1}

There are many places to read about continued fractions, including Chapter X of Hardy and Wright’s Intro. to the Theory of Numbers, §13.3 of Burton’s Elementary Number Theory, Chapter IV of Davenport, and Khintchine’s Continued Fractions. The notes

|x − c_{n}| = \frac{1}{q_{n} (\frac{1}{t_{n}} q_{n} + q_{n−1})}

< \frac{1}{q_{n}(a_{n+1}q_{n} + q_{n−1})}

= \frac{1}{q_{n} · q_{n+1}} ≤ \frac{1}{n(n + 1)} → 0.

Example. We use PARI to illustrate the convergence of the theorem for x = π.

? a = contfrac(Pi)
? [c[2]*1.0,c[4]*1.0,c[6]*1.0]   \\ even ones swoop down on pi.

1See http://www.research.att.com/~njas/sequences/DUNNE/TEMPERAMENT.HTML

CHAPTER 1. INFINITE CONTINUED FRACTIONS 4

Binary quadratic forms

Definition. The modular group SL_{2}(ℤ) is the group of all 2 × 2 integer matrices with determinant +1.

If g = ( \begin{matrix}p & q \\ r & s \end{matrix} ) ∈ SL_{2}(ℤ) and f(x, y) = ax^{2} + bxy + cy^{2} is a quadratic form, let

f|_{g}(x, y) = f(px + qy, rx + sy) = f ((\begin{matrix}p & q \\ r & s \end{matrix}) [\begin{matrix}x \\ y \end{matrix}]) ,

where for simplicity we will sometimes write f ([\begin{matrix}x \\ y \end{matrix}]) for f(x, y).

F = \{τ ∈ 𝔥 : Re(τ) ∈ [−\frac{1}{2}, \frac{1}{2}) , |τ| > 1 or |τ| = 1 and Re(τ) ≤ 0\} .

CHAPTER 2. BINARY QUADRATIC FORMS 6

Cyclotomic units

In the Chapter 2 we have defined quadratic forms.

Let p, q and r be different primes such that p, q, r ≡ 1 (mod 4). Let h denote the class number of ℚ(\sqrt{p}, \sqrt{q}, \sqrt{r}).

By σ_{l}, where l ∈ S, we denote the automorphism determined by the equation Gal (K_{S}/K_{S\{l}}) = {1, σ_{l}}.

Let us further define

ϵ_{n_{S}} = \begin{cases}1 & if S = ∅, \\ \frac{1}{\sqrt{l}} N_{ℚ^{S}/K_{S}}(1 − ζ_{S}) & if S = \{l\}, \\ N_{ℚ^{S}/K_{S}}(1 − ζ_{S}) & if #S > 1. \end{cases}

3.1 Finite sums

Below the usual way of calculating the sum of geometric series is shown:

1+q + q^{2} + ⋯ + q^{n}  = S,
q + q^{2} + ⋯ + q^{n}+q^{n+1} = qS  / · (−1)
1  −q^{n+1} = (1 − q) · S

3.1. FINITE SUMS  CHAPTER 3. CYCLOTOMIC UNITS

8

Study plans

1. rok studia

kód  název  kred. rozsah  učitel

Podzimní semestr Povinné předměty

M6531 Teorie množin 2+2 2/0 zk Fuchs, E.
M7521  Pravděpodobnost a statistika 1 4+2 2/2 zk Budíková, M.

Povinně volitelné předměty

M7531 Diplomová práce 4 0/0 z Šišma, P.

Jarní semestr Povinné předměty

M4520

Seminář ze středoškolské matematiky 2 2 0/2 k

Kučera, R.

M7511  Historie matematiky 1  2+1 2/0 kz Fuchs, E.
M8501  Didaktika matematiky 1  3 2/2 k Šimša, J.

Povinně volitelné předměty

M8532  Diplomová práce  4 0/0 z Šišma, P.

Index

continued fractions, 3 example, 3

PARI example, 3

quadratic form, 7 binary, 5–7

unit cyclotomic, 7