elementary number theory - Solutions for $x^2+y^2=2007$ and reference request.

I have never formally studied number theory, it is not a part of my course work, and what I have learnt is reading Wikipedia or the answers here. This question was on a test and I tried to use quadratic residues to solve this.

Find the number of solutions in integers to:

$$x^2+y^2=2007$$

We can observe that $2007\equiv 0\pmod3$

$x^2\equiv0,1\pmod{3}$, $y^2\equiv0,1\pmod{3}$

Since these can never add up to $3$,we conclude that $x$, $y$ are divisible by $3$

Let $x=3x'$, $y=3y'$

$$(x')^2+(y')^2=223$$

Since $223\equiv 3\pmod4$ but $x^2\equiv 0,1\pmod{4}$ we conclude there are no solutions.

Is this argument correct? Also how else could this question have been solved? (Considering the fact that we don't have modular arithmetic in our syllabus, and therefore the teacher was probably expecting something else)

Also where can I study number theory?

Answer

$$2007\equiv3\equiv-1\pmod4$$

But $\displaystyle a\equiv0,\pm1,2\pmod4\implies a^2\equiv0,1\pmod4$

So, what are possible values of $\displaystyle x^2+y^2\pmod 4$