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 squares.
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 squares in two ways.
Let
. . then: .
Example: .
We have: .
Then: .