Polynomial Detectors for Prime Powers via Divisor Sums

Description (Abstract): This record contains the scripts and experimental results associated with the preprint “Polynomial Detectors for Prime Powers via Divisor Sums” This work establishes a partition-theoretic framework for detecting prime powers $p^k$, proving that for fixed $k$, detectors arise from the Binomial Master Theorem $(M_1-1)^k = n(M_1-n)^k$. The study distinguishes the polynomial nature of prime power detection (in the arithmetic ring $\mathbb{Q}[n, M_1, M_2]$) from the linear Eisenstein framework used for primes. Key Contributions: Binomial Master Theorem: A generating theorem that yields explicit polynomial detectors for any fixed prime power $p^k$, with a rigorous geometric-series derivation. Universal & Hybrid Detection: A single Universal Polynomial $\mathcal{U}(n)$ that detects all prime powers simultaneously (verified for $k \le 26$), and hybrid detectors whose zero sets unify disjoint classes (e.g., primes, prime cubes, and semiprimes) via shared algebraic signatures. Prime Recovery Lemma: A proof that for any prime power $p^k$, the base prime $p$ is rationally determined by the pair $(n, M_1(n))$, reducing detection to polynomial elimination. Rigorous Verification: A dual verification strategy combining symbolic computer algebra (proving the identities vanish for all $p^k$) and large-scale numerical testing (checking the converse on $n \le 100,000,000$ with zero false positives). Files: Code `1_prime_power_detector_search.py`: The core discovery engine. Uses Linear Algebra (Nullspace analysis) to solve for polynomial dependencies in the MacMahon observable space. `2_merge_detectors.py`: The code merges the equations in files from running different iterations of detector code (for different powers, orders etc) into master detector files for easy input for verification script and readability etc. `3_verify_detectors.py`: The theorem prover. A script containing the hardcoded theorems (Universal Detector, Binomial Master Theorem) to verify them against integer sequences. 2. Verification Results `master_detectors_tuples.py`: A structured, Python-importable list containing every discovered polynomial equation as a tuple. `master_detectors_readable.txt`: A comprehensive, human-readable catalogue of all polynomial detectors discovered by the engine. `master_appendix.tex`: Auto-generated LaTeX code containing the discovered equations. 3. Paper Venkat2025_PrimePowerDetection_v4.pdf: The Version 4 preprint containing the Binomial Master Theorem, the Prime Recovery Lemma, and the complete derivation of the Universal and Hybrid detectors. License: Software files (.py): MIT License Data/Text files (.txt, .pdf): Creative Commons Attribution 4.0 International