Semi-primitive Roots and Irreducible Quadratic Forms

Modulo a prime number, we define semi-primitive roots as the square of primitive roots. We present a method for calculating primitive roots from quadratic residues, including semi-primitive roots. We then present progressions that generate primitive and semi-primitive roots, and deduce an algorithm to obtain the full set of primitive roots without any GCD calculation. Next, we present a method for determining irreducible quadratic forms with arbitrarily large conjectured asymptotic density of primes (after Shanks, 12). To this end, we propose an algorithm for calculating the square root modulo p, based on the Tonelli-Shanks algorithm 4.