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Panter · 14.08.2022, 23:05

Zsigmondy'S Theorem
MP4 | Video: h264, 1280x720 | Audio: AAC, 44.1 KHz
Language: English | Size: 10.90 GB | Duration: 8h 24m

Advancing Algebra to understand Number theory

What you'll learn
Proof of Zsigmondy's Theorem
Applications of Zsigmondy's Theorem
Lifting the Exponent Lemma
Cyclotomic polynomials
Complex roots of unity
Mobius Inversion
Number theory
Algebra
Requirements
Modular arithmetic with prime numbers
Sound algebra skills

Description
The story line that guides us is proving a theorem of Zsigmondy in number theory and seeing how it can be used to solve maths olympiad problems that would otherwise be quite difficult. To achieve this goal we first understand what I consider to be the most central topic in high school algebra which is omitted in high schools: cyclotomic polynomials. This sounds specialised but this is at the heart of all the algebra learned at high school such as factorising a difference of 2 squares or cubes. The cyclotomic polynomials gives a factorisation of x^n-1. When n is 2, this is just the difference of 2 squares. If you let the x be x/y then you really get x^2-y^2 after some easy manipulation. (x^n means x to the power of n)These lessons will be a very valuable part of a serious high school maths student or olympian.One of the really interesting features of this course is that the instructor learns the proof of the Zsigmondy Theorem with the students and you get to see how to educate yourself without further need to be taught.

Overview

Section 1: Introduction

Lecture 1 Introduction

Section 2: Polynomials

Lecture 2 PST polynomials part 1

Lecture 3 PST polynomials part 2

Lecture 4 Irreducibility

Section 3: Diophantine equations

Lecture 5 PST 3: Diophantine equations

Lecture 6 PST 3.11 Cyclotomic Recognition

Section 4: Mobius inversion

Lecture 7 Mobius inversion revisited

Section 5: Cyclotomic polynomials

Lecture 8 Cyclotomic polynomials skipping Mobius

Lecture 9 Cyclotomic polynomials including Mobius

Lecture 10 Cyclotomic recognition revisited

Lecture 11 Infinitely many primes 1 mod n

Lecture 12 2002 IMO Shortlist N3

Lecture 13 2006 IMO Shortlist N5

Section 6: LTE

Lecture 14 Summary summarised

Lecture 15 Problem 1

Lecture 16 2018 IMO Q4

Section 7: Zsigmond Prerequisites

Lecture 17 Prerequisites stated and some proved

Section 8: Zsigmondy Theorem Proof

Lecture 18 Relate to Cyclotomic polynomials

Lecture 19 Consolidate and continue

Lecture 20 psi_n=lambda_nP_n

Lecture 21 lambda_nP_n primes

Lecture 22 lambda_n=1 and other cases

Lecture 23 When a-b=1. QED

Section 9: Zsigmondy Applications

Lecture 24 a^n+b^n and other applications

Lecture 25 Problems

Maths olympiad students,Serious maths students,Serious students seeking proper foundation in algebra in high school

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