Semi-primitive roots and irreducible quadratic forms
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866916464079929344 |
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| author | Wolf, Marc Wolf, François |
| author_facet | Wolf, Marc Wolf, François |
| contents | 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]. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_20269 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Semi-primitive roots and irreducible quadratic forms Wolf, Marc Wolf, François General Mathematics 11A07 (Primary), 68P05, 11B05 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]. |
| title | Semi-primitive roots and irreducible quadratic forms |
| topic | General Mathematics 11A07 (Primary), 68P05, 11B05 |
| url | https://arxiv.org/abs/2407.20269 |