From the time of Fermat, there has been a long line of work that studies numbers that can be represented by quadratic forms.
What is interesting about this problem to number theorists?
To ask in another way, what does this problem do for number theory?
For instance, chinese remainder theorem can be used to solve any system of linear congruences in one variable. So I can appreciate the importance of this theorem in the theory of numbers. Quadratic reciprocity tell us if a solution to a quadratic congruence exists. Although, unlike the linear case where division algorithm allow us to construct reciprocals, the quadratic residue does not give a method to construct square roots (as far as I know).
I am comfortable with ring theory, basics of Galois theory and elementary number theory. So I would appreciate it if the answer explains the motivations in terms of what I know.