Is there a general method to expressing integers as the sum of two squares or do you just need to be good with numbers? For example, consider the following problem:
Express 605 as the sum of two squares
Now I believe that a solution to this problem exists because $605 = 5 \cdot 11^2$ and $11 \equiv 3 \pmod 4$ which appears an even number of times and $11$ is the only number in the prime factorization that is congruent to $3 \pmod 4$.
So looking at this problem, what are the first steps that you should take to try and solve this problem? Is there a quick way to go about this?