NTIC Related Questions About Sums

Section 14.3 Related Questions About Sums

There is yet another generalization that will serve better as a lead-in to the next chapters. Think about the following two problems.

What numbers can be written as \(x^2+2y^2\text{?}\) (Think of it as \(x^2+y^2+y^2\text{.}\))
What numbers can be written as \(x^2+3y^2\text{?}\)

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

Fact 14.3.1.

No number

\begin{equation*} n\equiv 5\text{ or }n\equiv 7\text{ (mod }8) \end{equation*}

can be written as \(x^2+2y^2\text{.}\)

Proof.

Try all numbers modulo 8 and see what is possible! (See Exercise 14.4.3.)

Already Fermat (unsurprisingly) claimed a partial converse to Fact 14.3.1. He stated that any prime number \(p\) which satisfies \(p\equiv 1\) 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

\begin{equation*} x^2+2y^2=(x-\sqrt{2}iy)(x+\sqrt{2}iy) \end{equation*}

you could start proving such things. When might a square root of two exist modulo \(p\) …

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

In Exercise 14.4.12, you will try to discover a similar pattern for \(x^2+3y^2\text{.}\) See also Section 15.4.