Minkowski to the Rescue

I had been struggling for several days last year to find a proof for the intriguing Fermat’s Two Square theorem which states the following :

If p is a prime number congruent to 1 mod 4, then it can be expressed as the sum of two squares.

At about the same time I had also been studying the Minkowski’s theorem for Convex Bodies from Hardy and Wright’s marvelous book on Number Theory : An Introduction to the Theory of Numbers (highly recommended). For some mysterious reason, this theorem stuck with me for quite a while — probably because it was my first introduction to the beautiful branch of Geometry of Numbers. 

Then one frosty Sunday night, I was marveling once again at the theorem when there was a sudden … connection. It was nothing short of magical; in the next 2 hours I worked through the details of how Minkowski’s theorem could be used to prove the Fermat’s two square theorem.

I have outlined my proof below.

The Minkowski’s theorem for Convex Bodies in two dimensions states the following :

Suppose L is a point lattice, the area of its fundamental parallelogram being A. Then, any convex region symmetrical about the origin O, and of area greater than 4A encloses at least one lattice point in addition to the origin.

F1

F2

I find this proof very pleasing due its sheer elegancy and its use of Minkowski’s theorem. And it is certainly always a treat to find a connection between two seemingly disparate entities!

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