Semi-primitive roots and irreducible quadratic forms

Marc Wolf; François Wolf

doi:10.29020/nybg.ejpam.v17i4.5372

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arXiv · 2407.20269

Semi-primitive roots and irreducible quadratic forms

Marc Wolf·François Wolf

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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].

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Marc Wolf, François Wolf. 2024-07-24. Semi-primitive roots and irreducible quadratic forms. https://doi.org/10.29020/nybg.ejpam.v17i4.5372

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