Semi-primitive Roots and Irreducible Quadratic Forms
DOI:
https://doi.org/10.29020/nybg.ejpam.v17i4.5372Keywords:
Primitive roots, semi-primitive roots, irreducible quadratic forms, asymptotic density, Fermat's theorem on sums of two squaresAbstract
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 [3].
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Copyright (c) 2024 Marc Wolf, François Wolf
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