Gaussian Integers and Fermat's two square theorem

I was trying to explain to someone recently what I thought mathematics was all about and why it interests me. I found it difficult to put into words some of the ideas and complexities I have seen over the last few years and I had trouble explaining why mathematics is such never-ending treasure chest of ideas. Sometimes you do get the odd glimpse of a fundamental principle and the topic I have just read about Gaussian integers seems to exemplify that.

What is a Gaussian integer? These are a generalization of the ordinary set of integers Z and can be defined as the set of complex numbers

$$ \mathbb{Z} \left[i\right]=\left\{a+bi:a,b\in  \mathbb{Z}\right\}$$

Is it possible to find similarities between this set and the set of ordinary integers and if so what do these common properties tell us about the nature of integers in general? Well it turns out that these two sets do have properties that are in common. For example, it is possible to show that there are Gaussian prime numbers much as there are ordinary prime numbers and you can go on to define properties such as division, greatest common divisor, unique prime factorisation etc which is all very interesting.

So how does this come about? Well, firstly you define the norm of an element of Z[i] to be

$$norm(a+bi)=|a+bi|^{2}=a^{2}+b^{2}$$

and from the properties of the modulus we know that

$$|z_{1}z_{2}|=|z_{1}||z_{2}|$$

and so norms multiply, that is

$$norm\left(z_{1}z_{2}\right)=norm\left(z_{1}\right)norm\left(z_{2}\right)$$

Now because the norm is a real natural number we can define a Gaussian prime to be a Gaussian integer that is not the product of Gaussian integers of smaller norm. For example, 3+2i is a Gaussian prime because norm(3+2i)=13 which is prime in Z and so not the product of Gaussian integers of smaller norm. On the other hand 3+i is not a Gaussian prime because norm(3+i)=10 and since 2 and 5 divide 10 then we might expect 3+i to be factorisable, which it is since 3+i=(1-i)(1+2i) and the norm of 1-i is 2 and that of 1+2i is 5.

When it comes to deciding whether ordinary primes are also Gaussian primes we come to an interesting property. For example, 2 is not a Gaussian prime because 2=(1+i)(1-i) and so can be divided by 1+i or 1-i. On the other hand 3 is a Gaussian prime. This is because norm(3)=9 and there are no Gaussian integers with norm 3 which divide 9 (the smallest non-trivial norms are 2, 5, 8...). Now 2 is the sum of two squares, whilst 3 is not and you can go on to show that ordinary prime p is a Gaussian prime if and only if p is not the sum of two squares (or alternatively ordinary prime p is the sum of two squares if and only if p is not a Gaussian prime).

This leads on to the proof of what I think is a fascinating theorem. Fermat's two square theorem states that if p=4n+1 is prime, then p=a²+b² for some a,b in Z. Since for p>2 every prime is either of the form 4n+1 or 4n+3, then this characterises half the primes as being the sum of two squares. The proof comes from the use of Lagrange's Lemma which states that a prime p=4n+1 divides m²+1 for some integer m (this comes from Wilson's theorem). Since m²+1=(m-i)(m+i) in Z[i] although p divides m²+1 it does not divide m-i or m+i as m/p - i/p and m/p + i/p are not Gaussian integers. By the Gaussian prime divisor property if a Gaussian prime divides the product of Gaussian numbers (m-i)(m+i) then it should divide either m-i or m+i and, as it doesn't, then p can't be a Gaussian prime. As we have seen from above this means that p is the sum of two squares.

Why I find this fascinating is that although prime numbers appear to be a random sequence of numbers we do have this structured form for half the set. It makes me wonder if there is any construction for primes of the form 4n+3? Also, are the a,b that are used to construct the primes of the form 4n+1 drawn from all of Z or are there numbers which are missing? All this gets me thinking!

AWOL

As you can see I haven't posted anything here for six months and some of you may be wondering what has happened. Well, to cut a long story short I abandoned my OU studies in February and, at present, I am not sure if I will be continuing on with them. Though I was reluctant to stop working on these courses, in the end I had to because they were taking up far too much of my time. On reflection it would have been better for me to have started just one of these courses instead of both of them and if I had I would probably have kept going. However, due to the changes that the OU made to the time limits for the degree I was taking and the fact that these courses were going to be discontinued, it wasn't possible for me to do this and so eventually I sort of gave in and didn't have the will power to continue. It is a pity but it isn't the end of the world. I feel it is a shame that the OU made these wholesale changes to their degree structure that left people like myself, who like to take things at a considered pace, high and dry. It's their loss as well as mine.

So what now? I have gone back to reading Elements of Number Theory by John Stillwell and have nearly got half way through the book. I am both reading and understanding the text as well as answering the exercises. It is a well written book and the exercises bring out what is in the text without being overly difficult. I have just finished the chapter on the Pell Equation and I liked the visual approach of Conway which is described. I also went through the previous chapter on the RSA cryptosystem and this enhances what we learnt at the OU. I have written some programs for coding and decoding messages on my other calculator blog.

As for the future I am not really sure what direction I should now take. I suspect I will continue on with the personal study until I get that desire to study more formally again. There are plenty of courses that are appearing online as Chris has been pointing out.

Slogging on

I am still slogging on with the number theory and groups courses. Regularly I ask myself why I am doing this, but then I know that I like having something to focus on that keeps me busy. But sometimes it is just so blooming difficult!

At the moment I am trying to cram in some work on TMA03 of the numbers course. I sort of dread it in advance because these TMAs are really quite difficult. Often the questions don't relate directly to the course material. They use the ideas but in a way that you haven't seen before, so it really makes you think. There is, however, the feeling of huge elation when you do manage to conquer a problem. I take these questions a small chunk at a time. Often I will read the next part of a question and start mulling it over in my head. At first it seems impossible, but after a few glints of ideas, you get an idea of the framework for an answer and then you can flesh the whole thing out. I often go to bed thinking about a knotty problem. It is a good time to give a problem your undevoted attention without distraction, but I can't say it is a good way to get to sleep!

I have now finished NT6 on quadratic reciprocity. One more book and I would have the third block finished for M381. The groups course is going ok. The problem is that the ideas are increasingly complicated and although I tenuously understand a book when I have read it, it rapidly gets forgotten after a few weeks. I have finished GE3 and am currently on GR4 which is investigating finite groups and what we know about them.