Empirical Verification of the Bateman-Horn Conjecture for the Degree-46 Polynomial Q₄₇(n) = n⁴⁷ - (n-1)⁴⁷
Year: 2026
Type: preprint
Abstract: We present a large-scale computational verification of the Bateman-Horn conjecture for the degree-46 polynomial Q₄₇(n) = n⁴⁷ - (n-1)⁴⁷ in the quadruplet configuration. By executing a systematic sieve over the range n ∈ [2.3 × 10⁷, 8 × 10¹¹], we discovered 2,359 prime quadruplets. Furthermore, using a targeted Chinese Remainder Theorem (CRT) search, we found eleven 1000-digit prime quadruplets at n ≈ 4.8 × 10²¹, all 44 constituent primes of which have been rigorously certified via Elliptic Curve Primality Proving (ECPP). We evaluate the Bateman-Horn structural constant C_BH ≈ 6514.2 utilizing an Euler product over 5.76 × 10⁶ primes, incorporating an Abel summation tail correction to handle conditional convergence. A standard χ² goodness-of-fit test comparing observed versus predicted counts across 12 data sectors yields χ² = 10.56 on 12 degrees of freedom (p ≈ 0.57), demonstrating excellent agreement with the asymptotic prediction.
Author: Chen, Ruqing
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7143334325
Cites:
Cited by:
Related to: