Zsigmondy's Theorem

Name: Zsigmondy's Theorem
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Advancing Algebra to understand Number theory

Created byDr Michael Sun

Last updated 5/2020

English

English [Auto],

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

Course content

9 sections • 25 lectures • 8h 25m total length

Introduction6:42
Concrete goals: Learn the proof of the Zsigmondy Theorem and apply it to number theory problems.
To do this we learn (LTE) Lifting the Exponent Lemma which is also useful for solving number theory problems.
We also learn about cyclotomic polynomials and general polynomial notions.
Abstract goals: Understand the point of cyclotomic polynomials and its role in algebra.
Use theoretical understanding as part of one's problem solving approach.
Schedule: 1 month for cyclotomic polynomials (1 week theory 2 weeks exercises 1 week solutions)
1 month for Zsigmondy (1 week for LTE plus exercises, 2 weeks for proof and 1 week for applications)
Improvements: Name the Z proof lectures add missing lecture on why PnLambdaN. Add PST 3. individual sections and upload scan of problems, Zapplication video,
To make things more efficient I'll schedule at least 1 improvement made with each student enrollment or prompted by student.

PST polynomials part 123:55
Critical reading with commentary of PST chapter on polynomials.
PST polynomials part 21:12:45
Irreducibility0:27
See linked video at 4 hour time stamp.
Here the concept of polynomials is gone more in depth and notions of irreducibility are defined and discussed.
Eisenstein's Criterion and Gauss' Lemma are proved and are helpful for this notion.
Cyclotomic polynomials are irreducible and it is not strictly necessary for the proof of Zsgimondy but knowing this still guides us. For example, Gauss' Lemma does come up in the proof and the lemma with double roots is essentially something that happens when the polynomials are irreducible.
If people are interested in going through a proof of the irreducibility of cyclotomic polynomials I will add it.
Author choices include: Gauss Dirichlet, and a couple more.

PST 3: Diophantine equations0:04
Some sections of PST 3 are here just to give you a refresher into basic number theory. The link in resources is time stamped, 3.7 and 3.8 towards end of video skipping geometry. See description for time stamp
We are live reading to give students the idea of how one can read a book critically.
PST 3.11 Cyclotomic Recognition39:40
Last section of diophantine equations chapter, a little unsatisfied with the punchline given the title involved cyclotomic polynomials, will resolve this later.

Cyclotomic polynomials skipping Mobius0:07
Live read of Cyclotomic polynomials notes skipping Mobius Inversion. Will go over everything again with Mobius inversion.
Cyclotomic polynomials including Mobius0:03
Check resources for video and notes. Here we assume Mobius inversion has been covered
Cyclotomic recognition revisited7:23
Infinitely many primes 1 mod n8:59
Explain a Dirichlet-inspired argument proving there are infinitely many primes 1 mod n by analyzing primes that divide a^k−1 and contradicting a finite list.
2002 IMO Shortlist N323:19
Make sure you try these problems yourself before watching these videos, the problems are on the worksheet or can be found separately online
2006 IMO Shortlist N530:42

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.

Who this course is for:

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

Zsigmondy's Theorem

What you'll learn

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Course content

Introduction1 lecture • 7min

Polynomials3 lectures • 1hr 37min

Diophantine equations2 lectures • 40min

Mobius inversion1 lecture • 46min

Cyclotomic polynomials6 lectures • 1hr 11min

LTE3 lectures • 1hr 12min

Zsigmond Prerequisites1 lecture • 44min

Zsigmondy Theorem Proof6 lectures • 2hr 9min

Zsigmondy Applications2 lectures • 1min

Requirements

Description

Who this course is for: