Hello, putnam120!

I happen to be familiar with this theorem,

. . but I don't know of a proof.

However, it also has an extra and amazing feature.

Prove that if two numbers can be written as the sum of two squares,

then the product of these numbers can also be written

as the sum of two squaresin two ways.

Let

. . then: .

Example: .

We have: .

Then: .