March 7

Plan:

• Questions
• Using the internet
• Using $\LaTeX$
• The Euclidean algorithm and the fundamental theorem of arithmetic

Student Questions: None

The discussion page can be found here, for observations and remarks.

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Class was less than half full. I'm still missing homework 3 from more than half the class. No one has posted any lecture notes for Tuesday's class. I don't know how to deal with dwindling participation. Just continue on with those who are paying attention and doing the work?

Using the internet: a good thing, however …

• You must credit your source (most people are doing this)
• You can't just cut something from wikipedia and paste it into your homework. Rule from now on: paste a chunk of what you've read (use $\LaTeX$ \begin{quotation} … \end{quotation}). Then rewrite the content in your own words, to prove to me and to yourself that the material has passed through your brain as well as through your computer.
• When you post a link on a discussion page, don't just say "this is cool". Provide some evidence that you read it, or parts of it. I don't expect you to rewrite it in your own words, but you should say something about what you learned, maybe what disappointed you too. Write a review so we know what to look for when we visit.

Using $\LaTeX$: it's required, both here on the wiki and in your homework. From now on that's all I will read.

Starting late, what we actually did:

The Euclidean algorithm, proving (constructively) that given integers $a$ and $b$ you can find integers $x$ and $y$ such that

(1)
\begin{align} ax+by = \text{gcd}(a,b). \end{align}

We explained this by working a representative numerical example ($a = 25, b = 89$). Of course the proper thing to do would be to find it in Euclid and read his proof.

Then we used the Euclidean algorithm to show that if a prime divides a product it divides one of the factors. It's an easy induction from there to the fundamental theorem of arithmetic.

Introduced the Gaussian integers $\{a+bi \mid a, b \text{ integers } \}$ and observed that $2 = (1+i)(1-i)$, $3$ is still prime, $5 = (2+i)(2-i)$. The generalization is Fermat's theorem saying that an odd prime is a sum of two squares if and only if it is congruent to 1 modulo 4.

Started work on the puzzle

(2)
\begin{align} 6 = 2 \times 3 = (1+ \sqrt{-5})(1 - \sqrt{-5}) \end{align}

and Kummer's invention of ideal complex numbers.