A Computational Framework for Large-Scale Prime Constellation Searches in Q(n) = n^p − (n−1)^p: Algorithms, Pre-Sieve Strategies, and Performance Engineering

We describe the complete computational framework underlying the Titan Project — a systematic census of prime constellations in the sequence Q(n) = n⁴⁷ − (n−1)⁴⁷ for n ≤ 2 × 10¹¹. The framework integrates three tiers of computation: (i) a segmented sieve that evaluates Q(n) modulo small primes to eliminate composite candidates before any large-integer arithmetic; (ii) gmpy2-accelerated Miller–Rabin primality testing with 25 rounds for numbers of 380–530 digits; and (iii) a satellite radar that scans the local prime landscape around each confirmed prime, exploiting the forbidden residue lattice to skip 33.3% of candidates without computation. The Residue Pre-Sieve (the "Titan Shield Filter") exploits the fixed congruence Q(n) ≡ 1 (mod 6) to eliminate all gap candidates k ≡ 1 (mod 3). We prove this filter is optimal for the mod-{2,3} lattice, derive the full multi-prime filter cascade (mod-3 through mod-43, achieving 3.39× theoretical speedup), and present the generalized filter for arbitrary prime exponents p in the family Q_p(n) = nᵖ − (n−1)ᵖ. Performance benchmarks on 500-digit numbers show that gmpy2.is_prime achieves ~15 ms per 25-round test, a factor of ~100× over native Python arithmetic. The full Titan census — scanning 2 × 10¹¹ values of n, confirming 742 quadruplets, 7 quintuplets, and 9,012 satellite primes — consumed approximately 1,200 CPU-hours. We address pseudoprime robustness on polynomial trajectories, proving that Carmichael number density at 500 digits (< 10⁻³³⁴) and the absence of algebraic pseudoprime bias in Q(n) make false positives negligible. The segmented search architecture with share-nothing log file synchronization enables embarrassingly parallel deployment on multi-core and distributed systems. The framework is designed for portability: replacing the exponent p = 47 with any odd prime yields a complete constellation search pipeline requiring only trivial parameter changes. A recipe for new exponents is provided, covering resonant prime classification, pre-sieve construction, yield estimation via Bateman–Horn, and the three-tier pipeline. Paper contents: 15 pages, 1 figure (pipeline flowchart), 8 tables, 2 algorithms, 2 code listings, 13 references. Related repositories: • Part IV: https://github.com/Ruqing1963/Q47-Satellite-Primes • Part II: https://github.com/Ruqing1963/Q47-Deep-Space-Quadruplet-Census • Part III: https://github.com/Ruqing1963/Q47-Quintuplet-Sextuplet-Boundary