Quantitative Predictions for Prime Values of the Titan Polynomial: The Bateman--Horn Constant and Asymptotic Density
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| Natura: | Recurso digital |
| Lingua: | inglese |
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Zenodo
2026
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| _version_ | 1866901598846844928 |
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| author | Chen, Ruqing |
| author_facet | Chen, Ruqing |
| contents | <p>I compute the Bateman–Horn constant C_Q for the Titan polynomial Q(n) = n⁴⁷ − (n−1)⁴⁷, a degree-46 cyclotomic norm form whose values have no prime factor less than 283. The 60 primes below 283 each contribute a factor p/(p−1) > 1 to the Euler product, yielding a small-primes boost P_small ≈ 10.19 (consistent with the Mertens approximation e^γ ln(283) ≈ 10.05). The splitting primes p ≡ 1 (mod 47), starting at p = 283, contribute suppression factors (1−46/p)/(1−1/p) < 1. The net product converges to C_Q ≈ 8.68.</p> <p>Under the Bateman–Horn heuristic, the prime counting function satisfies π_Q(x) ~ (C_Q/46) Li(x) ≈ 0.1887 Li(x), meaning prime values of Q(n) occur with a frequency approximately 8.7 times that predicted for a generic degree-46 polynomial. Direct computation confirms π_Q(10,000) = 232 vs predicted 235 (error −1.3%) and π_Q(20,000) = 429 vs predicted 432 (error +0.6%), consistent with the asymptotic prediction.</p> <p>The repository includes the paper (LaTeX source and compiled PDF, 3 pages), three CSV data files (Euler product local factors for 110 primes, convergence of C_Q at five truncation limits up to 10⁷, and observed vs predicted prime counts at six checkpoints up to x = 20,000), and three Python verification scripts. All scripts produce correct results.</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_18526470 |
| institution | Zenodo |
| language | eng |
| publishDate | 2026 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | Quantitative Predictions for Prime Values of the Titan Polynomial: The Bateman--Horn Constant and Asymptotic Density Chen, Ruqing Bateman-Horn conjecture cyclotomic norm form Euler product polynomial primes shielding property prime density local root count Mertens theorem splitting primes asymptotic prediction Mathematics Number Theory <p>I compute the Bateman–Horn constant C_Q for the Titan polynomial Q(n) = n⁴⁷ − (n−1)⁴⁷, a degree-46 cyclotomic norm form whose values have no prime factor less than 283. The 60 primes below 283 each contribute a factor p/(p−1) > 1 to the Euler product, yielding a small-primes boost P_small ≈ 10.19 (consistent with the Mertens approximation e^γ ln(283) ≈ 10.05). The splitting primes p ≡ 1 (mod 47), starting at p = 283, contribute suppression factors (1−46/p)/(1−1/p) < 1. The net product converges to C_Q ≈ 8.68.</p> <p>Under the Bateman–Horn heuristic, the prime counting function satisfies π_Q(x) ~ (C_Q/46) Li(x) ≈ 0.1887 Li(x), meaning prime values of Q(n) occur with a frequency approximately 8.7 times that predicted for a generic degree-46 polynomial. Direct computation confirms π_Q(10,000) = 232 vs predicted 235 (error −1.3%) and π_Q(20,000) = 429 vs predicted 432 (error +0.6%), consistent with the asymptotic prediction.</p> <p>The repository includes the paper (LaTeX source and compiled PDF, 3 pages), three CSV data files (Euler product local factors for 110 primes, convergence of C_Q at five truncation limits up to 10⁷, and observed vs predicted prime counts at six checkpoints up to x = 20,000), and three Python verification scripts. All scripts produce correct results.</p> |
| title | Quantitative Predictions for Prime Values of the Titan Polynomial: The Bateman--Horn Constant and Asymptotic Density |
| topic | Bateman-Horn conjecture cyclotomic norm form Euler product polynomial primes shielding property prime density local root count Mertens theorem splitting primes asymptotic prediction Mathematics Number Theory |
| url | https://doi.org/10.5281/zenodo.18526470 |