Recently i was researching about QRs and i started to think about how we can utilize them. First i asked myself what are the QRs and NRs under the hood. I came up with the following definition: A congruence class A is QR for modulus P IFF
$$xP + A = y^2$$
for some integers $x, y$. So if we rearrange the equation above we can obtain the following form,
$$y^2 -Px = A.$$
which is indeed a Quadratic Diophantine Equation. Are Quadratic Residues the way to the solutions of Quadratic Diophantine Equations? Or for showing them indeed they have solutions?