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After a couple of people have asked for it, I thought I’d upload my Yeats Prize essay, written during the summer of 2010 on the subject of Sheaf Theory. Enjoy!

Sheaftheory

It is a classical result of number theory that every prime number $p \equiv 1 \text{ mod }4$ is a sum of two squares. I stumbled across the following absurd proof of this fact at http://www.jstor.org/stable/2323918, apparently due to Prof Roger Heath-Brown of Oxford.

Consider the set $S = \{(x,y,z) \in \mathbb{N}^3 : x^2+4yz = p\}$. This is clearly finite and nonempty for $p \equiv 1 \text{ mod } 4$.

Apply to it the following map:

$(x,y,z) \mapsto \begin{cases} (x+2z, z,y-x-z) \text{ if } x2y \end{cases}$

One can check that this defines a map from $S$ to $S$, and furthermore that it is an involution (applying it twice gives the identity), with the unique fixed point $(1,1,\frac{p-1}{4})$, so in particular $S$ is odd, so the involution $(x,y,z) \mapsto (x,z,y)$ must also have a fixed point, giving a value $y$ with $x^2 + (2y)^2 = p$.

I’m really not sure why this works. I suppose our crazy involution looks slightly like the kind of thing appearing in the theory of reduction of quadratic forms…. so maybe some connection with modular actions…. I don’t know… perhaps, like all good magic, I don’t yet want to see how this trick is done until I’ve resolved to attempt to perform it myself.

However, I guess if anyone preparing for the part IB “Groups Rings and Modules” questions feels like writing something outrageous in an exam, this might be useful.