Semi-primitive roots and irreducible quadratic forms

TL;DR

This paper introduces methods for calculating primitive and semi-primitive roots modulo primes, and presents algorithms for identifying irreducible quadratic forms and computing modular square roots, advancing number theory techniques.

Contribution

It provides new algorithms for generating primitive and semi-primitive roots and for identifying irreducible quadratic forms with high prime density.

Findings

Algorithm for generating primitive roots without GCD calculations

Method for identifying irreducible quadratic forms with high prime density

Implementation of Tonelli-Shanks algorithm for modular square roots

Abstract

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, [1][2]). To this end, we propose an algorithm for calculating the square root modulo p, based on the Tonelli-Shanks algorithm [4].