It is a classical result of number theory that every prime number 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 . This is clearly finite and nonempty for .

Apply to it the following map:

One can check that this defines a map from to , and furthermore that it is an involution (applying it twice gives the identity), with the unique fixed point , so in particular is odd, so the involution must also have a fixed point, giving a value with .

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.

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April 12, 2011 at 10:40 pm

James CranchI think in this form it is not due to Heath-Brown, but to Don Zagier: it was, however, based on an earlier proof by DRH-B.

April 13, 2011 at 10:29 am

tloveringAh, sure. That’s indeed the paper I referenced above, of which I clearly didn’t really read the blurb very carefully.