On primitive solutions of the Diophantine equation \(x^2 + y^2 = M\) (Q2669024)

This is an elementary discussion concerning explicit formulae for the primitive integer solutions of the equation \(x^2+y^2=M\) (where \(M\) is a product of primes of the form \(4n+1\)) using the Gaussian integers. Although this method can be considered folklore, it is indeed surprising that such a discussion does not seem to have been written down earlier. Therefore, it is a useful addition to the abundant literature on sums of two squares.