Statistical Morphology and Geodesic Rigidity of Prime Constellations in Q(n) = n⁴⁷ − (n−1)⁴⁷: A Complete Census of the First 2 Billion Cases
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| Format: | Recurso digital |
| Langue: | anglais |
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Zenodo
2026
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| _version_ | 1866901477159600128 |
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| author | Chen, Ruqing |
| author_facet | Chen, Ruqing |
| contents | <p>We present an exhaustive morphological census of prime-generating integers n for the polynomial Q(n) = n⁴⁷ − (n−1)⁴⁷ over the continuous domain n ∈ [1, 2×10⁹]. The survey identifies 18,473,571 probable primes classified into four morphological types: 18,121,562 solitary primes, 173,351 pairs, 1,749 triplets, and 15 quadruplets (four consecutive integers each generating a probable prime of 341–430 digits).</p> <p>Key results:</p> <p>(1) Discovery of a 15th quadruplet at n = 23,159,557, previously absent from all published catalogs, recovered by filling the [10⁶, 10⁸] data gap.</p> <p>(2) Modular-3 Exclusion Principle: We prove that Q(n) ≡ 1 (mod 3) universally, rendering Q(n)+k composite for every offset k ≡ 2 (mod 3). This eliminates one-third of all candidate positions — including the right twin-prime slot k = +2 — establishing an intrinsic left–right chirality in the satellite field.</p> <p>(3) Geodesic Rigidity: The pair-to-solitary ratio R₂(x) decays only ~7% over [0.5×10⁹, 2×10⁹], significantly slower than the ~11% predicted by random independence models, confirming that constellation structure is arithmetically protected.</p> <p>(4) Asymptotic Validation: The observed ratio decay matches the Bateman–Horn prediction R(n) ~ K/ln(n) to within 1 percentage point (14% predicted vs. 15% observed over a factor-of-20 range).</p> <p>(5) Conditional Neighborhood Sieve: A formally defined search space S = {Q(n)−k | n ∈ C₄, k ∈ [2,R], k ≢ 2 (mod 3)} that exploits chirality and conditional density for deep-space surveys beyond n = 10¹¹.</p> <p>This repository contains the full paper (LaTeX source + compiled PDF), three publication-quality figures, the complete per-bin census data, the 15-quadruplet catalog, and the Python algorithms (exhaustive multi-core sweeper and chirality-aware deep-space radar) used in the survey.</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_18701355 |
| institution | Zenodo |
| language | eng |
| publishDate | 2026 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | Statistical Morphology and Geodesic Rigidity of Prime Constellations in Q(n) = n⁴⁷ − (n−1)⁴⁷: A Complete Census of the First 2 Billion Cases Chen, Ruqing prime constellations polynomial primes consecutive prime generators Bateman–Horn conjecture Hardy–Littlewood conjecture prime quadruplets geodesic rigidity arithmetic geometry primality testing Miller–Rabin computational number theory Q(n) = n⁴⁷ − (n−1)⁴⁷ Number Theory Computational Mathematics Mathematical Software <p>We present an exhaustive morphological census of prime-generating integers n for the polynomial Q(n) = n⁴⁷ − (n−1)⁴⁷ over the continuous domain n ∈ [1, 2×10⁹]. The survey identifies 18,473,571 probable primes classified into four morphological types: 18,121,562 solitary primes, 173,351 pairs, 1,749 triplets, and 15 quadruplets (four consecutive integers each generating a probable prime of 341–430 digits).</p> <p>Key results:</p> <p>(1) Discovery of a 15th quadruplet at n = 23,159,557, previously absent from all published catalogs, recovered by filling the [10⁶, 10⁸] data gap.</p> <p>(2) Modular-3 Exclusion Principle: We prove that Q(n) ≡ 1 (mod 3) universally, rendering Q(n)+k composite for every offset k ≡ 2 (mod 3). This eliminates one-third of all candidate positions — including the right twin-prime slot k = +2 — establishing an intrinsic left–right chirality in the satellite field.</p> <p>(3) Geodesic Rigidity: The pair-to-solitary ratio R₂(x) decays only ~7% over [0.5×10⁹, 2×10⁹], significantly slower than the ~11% predicted by random independence models, confirming that constellation structure is arithmetically protected.</p> <p>(4) Asymptotic Validation: The observed ratio decay matches the Bateman–Horn prediction R(n) ~ K/ln(n) to within 1 percentage point (14% predicted vs. 15% observed over a factor-of-20 range).</p> <p>(5) Conditional Neighborhood Sieve: A formally defined search space S = {Q(n)−k | n ∈ C₄, k ∈ [2,R], k ≢ 2 (mod 3)} that exploits chirality and conditional density for deep-space surveys beyond n = 10¹¹.</p> <p>This repository contains the full paper (LaTeX source + compiled PDF), three publication-quality figures, the complete per-bin census data, the 15-quadruplet catalog, and the Python algorithms (exhaustive multi-core sweeper and chirality-aware deep-space radar) used in the survey.</p> |
| title | Statistical Morphology and Geodesic Rigidity of Prime Constellations in Q(n) = n⁴⁷ − (n−1)⁴⁷: A Complete Census of the First 2 Billion Cases |
| topic | prime constellations polynomial primes consecutive prime generators Bateman–Horn conjecture Hardy–Littlewood conjecture prime quadruplets geodesic rigidity arithmetic geometry primality testing Miller–Rabin computational number theory Q(n) = n⁴⁷ − (n−1)⁴⁷ Number Theory Computational Mathematics Mathematical Software |
| url | https://doi.org/10.5281/zenodo.18701355 |