As a lead-in to the next set of topics, think about the following two problems.

• What numbers can be written as \(x^2+2y^2\)?
• What numbers can be written as \(x^2+3y^2\)?

These are very natural generalizations to the “two squares” question. How could we approach them? Here's one approach.

Already Fermat (unsurprisingly) claimed a partial converse. He stated that any prime \(p\) which is \(p\equiv 1\) or \(p\equiv 3\text{ mod }(8)\) could be written as a sum of a square and twice a square.

This time, Euler wasn't the one who proved it! But you could almost imagine that by factoring \[x^2+2y^2=(x-\sqrt{2}iy)(x+\sqrt{2}iy)\] you could start proving such things.

Here are some numbers which can be written in this form.

In the exercises, you will try to discover a similar pattern for \(x^2+3y^2\).