MATH 25 CLASS 7 NOTES, OCT 5 2011
Contents
1. On the number of primes
1
Quick links to denitions/theorems
Euclids Theorem
Dirichlets Theorem
Green-Tao Theorem
Prime number theorem
1. On the number of primes
Say you write down the rst few primes: 2, 3,
MATH 25 CLASS 8 NOTES, OCT 7 2011
Contents
1. Prime number races
2. Special kinds of prime numbers: Fermat and Mersenne numbers
3. Fermat numbers
1
2
3
1. Prime number races
We proved that there were innitely many primes of the form 4k + 3, and we said
th
MATH 25 CLASS 28 NOTES, NOV 28 2011
Contents
1. Evaluating
1
p
1
2
2. Gauss Lemma, evaluating p
3. The quadratic reciprocity law: statement
1. Evaluating
We begin by evaluating
1
p
2
4
1
p
. Fortunately, we have done most of the work already.
Proposition
MATH 25 CLASS 27 NOTES, NOV 22 2011
Contents
1. Quadratic congruences: Introduction
2. When n = p: introduction and notation
1
2
1. Quadratic congruences: Introduction
We conclude this class by considering the analogue of a classical problem from
secondar
MATH 25 CLASS 26 NOTES, NOV 21 2011
Contents
1. Calculations involving primitive roots
1
1. Calculations involving primitive roots
Lets look at a few concrete calculations involving primitive roots. First, lets
consider the question of nding primitive roo
MATH 25 CLASS 25 NOTES, NOV 18 2011
Contents
1. The group U2e
2. The general case: Un
1
2
1. The group U2e
We have shown that Upe is cyclic for any prime power pe where p is odd. In contrast,
we will see that U2e is not cyclic, for e 3, but we will be abl
MATH 25 CLASS 24 NOTES, NOV 16 2011
Contents
1. Finding primitive roots in Up2
2. Finding primitive roots in Upe , p odd
1
1
1. Finding primitive roots in Up2
In the previous class, we saw that Up is cyclic, and so has primitive roots. We now
want to show
MATH 25 CLASS 23 NOTES, NOV 14 2011
Contents
1. Testing for primitive roots
2. Up is cyclic
1
2
1. Testing for primitive roots
The central question we want to answer right now is the following: when is Un
cyclic? If Un is cyclic, we call any g mod n (whic
MATH 25 CLASS 21 NOTES, NOV 7 2011
Contents
1. Groups: denition
2. Subgroups
3. Isomorphisms
1
2
4
1. Groups: definition
Even though we have been learning number theory without using any other parts of
mathematics, in secret we have been doing a lot of ab
MATH 25 CLASS 20 NOTES, NOV 4 2011
Contents
1. Solving polynomial congruences to prime power moduli
2. Hensels Lemma
1
3
1. Solving polynomial congruences to prime power moduli
Right now we still have no better way to solve f (x) 0 mod p than brute force.
MATH 25 CLASS 19 NOTES, NOV 2 2011
Contents
1. An old cryptosystem: the Caesar cipher
2. A modern cryptosystem: RSA
1
3
1. An old cryptosystem: the Caesar cipher
We will take a brief detour to learn about a simple and elegant, but important,
application o
MATH 25 CLASS 9 NOTES, OCT 10 2011
Contents
1. A short introduction to primality testing and factorization
2. Generating lists of prime numbers: the sieve of Eratosthenes
1
4
1. A short introduction to primality testing and factorization
When studying pri
MATH 25 CLASS 10 NOTES, OCT 12 2011
Contents
1. Congruences: introduction
1
1. Congruences: introduction
Number theory asks questions about integers, such as when certain equations have
solutions in integers. When studying these sorts of problems, it some
MATH 25 CLASS 11 NOTES, OCT 14 2011
Contents
1. More on congruences
2. Linear equations mod n
1
2
1. More on congruences
It is often useful to know that a single congruence mod n can be split up into
several congruences mod prime powers, and vice versa, m
MATH 25 CLASS 5 NOTES, SEP 30 2011
Contents
1. Prime numbers and prime factorization
1
Quick links to denitions/theorems
The Fundamental Theorem of Arithmetic
1. Prime numbers and prime factorization
Weve spent a good amount of time discussing divisibili
MATH 25 CLASS 5 NOTES, SEP 30 2011
Contents
1. A brief diversion: relatively prime numbers
2. Least common multiples
3. Finding all solutions to ax + by = c
1
3
4
Quick links to denitions/theorems
Euclids Lemma (important!)
1. A brief diversion: relative
MATH 25 CLASS 4 NOTES, SEP 28 2011
Contents
1. Bezouts Identity
1
Quick links to denitions/theorems
The main theorem on solving a linear equation in integers
1. Bezouts Identity
It turns out that the Euclidean algorithm can help us solve other problems r
MATH 25 CLASS 3 NOTES, SEP 26 2011
Contents
1. Euclidean division
2. Greatest common divisor
3. The Euclidean algorithm for calculating gcds
1
2
3
Quick links to denitions/theorems
Uniqueness, existence of Euclidean division
Denition of common divisor,
MATH 25 CLASS 2 NOTES, SEP 23 2011
Contents
1.
2.
3.
4.
Set notation
Logical statements
Proof by induction
Divisibility
1
2
3
4
Quick links to denitions/theorems
Set denition
The method of induction
Denition of divisibility
1. Set notation
Before delvi
MATH 25 CLASS 1 NOTES, SEPTEMBER 21 2011
1. A very brief and vague introduction to number theory
Whats number theory about? How is it dierent from, say, calculus or linear algebra,
which are two staples of an introductory college mathematics curriculum? I
MATH 25 CLASS 16 NOTES, OCT 26 2011
Contents
1. Fast exponentiation mod n
2. Fermats Little Theorem as a compositeness test
3. Strong psuedoprimes and the Miller-Rabin test
1
2
4
1. Fast exponentiation mod n
Today we will discuss how one might use Fermats
MATH 25 CLASS 15 NOTES, OCT 24 2011
Contents
1. Polynomial congruences mod p
2. Fermats Little Theorem
1
2
1. Polynomial congruences mod p
Lets think about polynomial congruences mod p, where p is a prime. The previous
proposition, which considered x2 1 m
MATH 25 CLASS 13 NOTES, OCT 19 2011
Contents
1. Polynomial congruences
1
1. Polynomial congruences
We now have a good understanding of how to solve systems of linear congruences
to dierent moduli, regardless of whether they are mutually coprime or not. In
MATH 25 CLASS 12 NOTES, OCT 17 2011
Contents
1. Simultaneous linear congruences
2. Simultaneous linear congruences
1
2
1. Simultaneous linear congruences
There is a story (probably apocryphal) about how certain generals from ancient
China would count thei
MATH 25 CLASS 16 NOTES, OCT 26 2011
Contents
1. A generalization of Fermats Little Theorem
2. Calculating (n)
1
2
1. A generalization of Fermats Little Theorem
Weve gotten a lot of mileage out of Fermats Little Theorem. It says that if p is
prime, and p a