Sum of Two Squares: Intro to Number Theory

[matqkks]

Why bother writing a given integer as the sum of two squares? Does this have any practical application? Is there an introduction on a first year number theory course which would motivate students to study the conversion of a given integer to sums of two squares?

 

[certainly]

There is a good deal of material. For instance the corollary to a theorem of Fermat :- "Any prime $p$ of the form $4k+1$ can be represented uniquely as the sum of 2 squares. Or (related) Lagrange's theorem:- "Any positive integer $n$ can be written as the sum of 4 squares, some of which may be 0".
Then there are also many interesting properties of $r_2(n)$ where $r$ is the sum of squares function.
I would recommend you research "representations of integers as sums of squares."

 

[micromass]

Why bother writing a given integer as the sum of two squares? Does this have any practical application?

It most likely does not. There are applications of number theory, but overall you should take the class mainly because you find it interesting, not because of possible applications.