So far, we have determined at least when some quadratic congruences have solutions, but most cases still seem hopeless. We certainly won't want to use Euler's Criterion directly for every single one.

Yet one might think we have to keep doing this, and that our task of finding out when numbers have square roots mod (\(p\)) is hopeless! But quite the opposite is true; in fact, we will derive results that took quite a bit of work almost effortlessly using the great theorem called quadratic reciprocity!