[yantrams]

I came across the Brahmagupta's identity mentioned here recently and thought it was pretty cool. https://en.wikipedia.org/wiki/Brahmagupta%27s_identity

It says - Numbers of the form a^2 + n*b^2 are closed under multiplication.

[ogogmad]

It follows from matrices of the form [[a, (-n)b], [b, a]] being closed under multiplication, and taking determinants.

In more advanced language: For R a commutative ring (like say, the integers) the following function f is a ring homomorphism

  f:R[√(-n)] -> M_2(R),
  f(a + b√(-n)) = [[a, (-n)b], [b, a]]

Now take determinants.

[madcaptenor]

There's a nice characterization of sums of two squares in terms of their prime factorization), namely that all primes of form 4k+3 appear with even multiplicity. From a quick look through OEIS it looks like there are similar characterizations for a^2 + n*b^2 but this is where I tap out on number theory.