# Chapter 4 of 24 - Fermat's and Wilson's Theorems

Okay. Another chapter down. Very much "congruence continued".

Sections I covered were Fermat's Little Theorem, representation of fractions by decimals, Wilson's Theorem, and polynomial congruences. On top of that, I've managed to complete my first assignment based on these first four chapters. Reviewing that feedback will be very interesting.

One thing I have noticed is that the materials in the book are far more challenging than the questions in the assignment. I wonder if I'm being lured in to a false sense of security here. I suppose I'll only know when I review the past exam papers for this module...

Can't quite believe I've reached this point as the next chapter is "Examples of groups". Hello group theory my old friend!

# Chapter 3 of 24 - Congruence

I've worked with congruences before, but this chapter was a lot more fun than I thought it would be. Introduction to properties, classic divisibility tests and simultaneous linear congruences.

Questions in this chapter seemed a lot more achievable. Though looking at the assignment, the related question seemed completely out of my reach. It was that feeling of fear you normally get in an exam when you see a question you don't understand and your brain shuts down. I've got some work to do in this area, as I feel I've been experiencing that feeling fairly consistently in the last two exams I've had (stats and complex analysis). -both of which have been covid lockdown at-home exams.

What was good though was in this chapter there was some "highest common factor" questions thrown in there, so that was really useful. Good for building confidence.

It seems there's more congruence work in the next section, "Fermat's and Wilson's Theorems".

# Chapter 2 of 24 - Prime Numbers

Right. Another very tough chapter complete.

Well I say complete. I did have to leave out a number of the exercises again. Of the answers to the exercises that I understand, the answers in most cases are inspired. There's simply no way I could ever make the logical bounds demonstrated in those answers.

I was so disturbed by the difficulty of the exercises in Chapter 1, I decided to consult my tutor about it. He replied explaining that a lot of the exercises are meant to teach you by you looking at the answers so you're kind of meant to get stuck. This made me feel a lot better, but now it's just down to the time I can spend reviewing and learning from each answer.

But yes. Prime numbers. Fun chapter! Covered an intro to The Primes, the prime decomposition of integers, the infinitude of primes, famous problems concerning primes, and Fibonacci numbers. Fibonacci numbers are far more interesting than I ever realised, it turns out.

Okay. Time to move on to Congruence...

# Chapter 1 of 24 - Foundations

Well 2021 has gone quickly hasn't it?

The workload in complex analysis became so large that I couldn't contribute to this blog for the rest of the academic year. This is frustrating as I felt there were some really important things to learn from my marked assignments. If I ever find time to write them up, I will do.

Though needless to say, it's suddenly October. My new (and final) module begins! M303 Further Pure Mathematics. This will be a big one. It's a final-stage double-credit module and contains an absolutely enormous amount of material.

Last week I received an email from my tutor that could be summarised to two words: "GET AHEAD". Of course that's when the fear struck, so I decided to be tactical with the first chapter. There were two sets of exercises I didn't attempt, because I realised I knew enough about the material to answer the assignment questions. However practically all of the exercises I did try, I couldn't complete. At this early stage, I'm perhaps feeling it might've been a mistake to choose this module. Having said that, I have been able to answer the first two questions of the assignment.

The first chapter was about number theory. Proof by Mathematical Induction, highest common factors, lowest common multiples, the Euclidean Algorithm, and Diophantine equations. Not only all that, but all the theories, definitions, propositions and lemmas that go along with them.

I feel this was so difficult because you need to use all the theories and definitions like a palette of different paints. In the same way as the artist it feels like the mathematician needs to use the theories and definitions to paint a picture. Problem being, it felt like I was being taught what the primary colours were for the first time. Still, we've moved on, and I've marked the exercises I need to return to. I'm hoping that down the line I'll be able to return to them and they'll make a bit more sense to me.

Deep breaths.