wonderings

suppose n, a, and b are integers, a and b > 0, a != b, and n = a^2 + b^2

sqrt(n) = sqrt(a^2 + b^2)
sqrt(n) = ||a + bi|| by the definition of the complex modulus
n = ||(a + bi)^2|| by the fact that the square of a complex number has a modulus equal to the square of the modulus of the original number
n = ||a^2 - b^2 + 2abi||
n = sqrt((a^2 - b^2)^2 + (2ab)^2) definition of complex modulus
n^2 = (a^2 - b^2)^2 + (2ab)^2

conclusion: any number that can be written as the sum of two squares has a square that can also be written as the sum of two squares. by induction, this holds for all higher n^2^k, so long as the corresponding a's and b's are never equal and never equal zero.