Quantitative Predictions for Prime Values of the Titan Polynomial: The Bateman--Horn Constant and Asymptotic Density

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.

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.

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.