Arithmetic Structure of the Sapri Aurea Denominators
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| Natura: | Recurso digital |
| Lingua: | En |
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2026
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| author | Ettore, Bevilacqua |
| author_facet | Ettore, Bevilacqua |
| contents | <p>This paper investigates the denominators D_n appearing in the power series expansion of the function K_φ(k) = (π/2)·((1+√(1-k²))/2)^(1/φ), where φ = (1+√5)/2 is the golden ratio.</p> <p>Using exact arithmetic in Q(√5), we compute D_n for n ≤ 35 and discover a remarkable arithmetic structure: the odd part of D_n contains only primes p ≡ ±2 mod 5 (the inert primes in the ring Z[φ]), and each such prime appears for the first time exactly at n = p.</p> <p>The result parallels the classical von Staudt–Clausen theorem for Bernoulli numbers, but restricted to the inert primes. It reveals a deep connection between the golden ratio, hypergeometric functions, and the arithmetic of quadratic fields.</p> <p>**Code availability:** All computations are reproducible. The Python implementation is available at:<br>https://github.com/SapriZero/SapriAurea/tree/main/code/python/denominators.py</p> <p>**Keywords:** elliptic integrals, golden ratio, hypergeometric functions, Bernoulli numbers, prime numbers, quadratic fields, Z[φ]</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_19188552 |
| institution | Zenodo |
| language | enc |
| publishDate | 2026 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | Arithmetic Structure of the Sapri Aurea Denominators Ettore, Bevilacqua <p>This paper investigates the denominators D_n appearing in the power series expansion of the function K_φ(k) = (π/2)·((1+√(1-k²))/2)^(1/φ), where φ = (1+√5)/2 is the golden ratio.</p> <p>Using exact arithmetic in Q(√5), we compute D_n for n ≤ 35 and discover a remarkable arithmetic structure: the odd part of D_n contains only primes p ≡ ±2 mod 5 (the inert primes in the ring Z[φ]), and each such prime appears for the first time exactly at n = p.</p> <p>The result parallels the classical von Staudt–Clausen theorem for Bernoulli numbers, but restricted to the inert primes. It reveals a deep connection between the golden ratio, hypergeometric functions, and the arithmetic of quadratic fields.</p> <p>**Code availability:** All computations are reproducible. The Python implementation is available at:<br>https://github.com/SapriZero/SapriAurea/tree/main/code/python/denominators.py</p> <p>**Keywords:** elliptic integrals, golden ratio, hypergeometric functions, Bernoulli numbers, prime numbers, quadratic fields, Z[φ]</p> |
| title | Arithmetic Structure of the Sapri Aurea Denominators |
| url | https://doi.org/10.5281/zenodo.19188552 |