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
Homepage

https://www.udemy.com/course/zsigmondys-theorem

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