Unified p-Adic Bounds and Computation for the Erdős–Moser Equation

This deposit presents the latest version (v3.4) of “Mahler Valuations and Verified Computation for the Erdős–Moser Equation.” We introduce a closed-form $p$-adic valuation formula for odd-prime moduli and integrate it with a novel Mahler-interpolation lemma for even exponents. Together, these yield a unified and general framework for bounding the $p$-adic valuation of power sums:

∑a=1p−1ak\sum_{a=1}^{p−1} a^ka=1∑p−1ak

for all exponents $k \ge 2$, bridging classical Lifting-The-Exponent (LTE) arguments with interpolation-theoretic tools.

The paper rigorously verifies these bounds across a wide computational range. For all odd primes $3 \le p < 200$ and even exponents $2 \le k \le 20$, no counterexamples to the proposed bounds are found. Additionally, a full near-miss survey is conducted for the Erdős–Moser ratio:

ρ(m,k)=∑i=1m−1ikmk\rho(m,k) = \frac{\sum_{i=1}^{m−1} i^k}{m^k}ρ(m,k)=mk∑i=1m−1ik

over the domain $2 \le m \le 2000$, $2 \le k \le 12$. No new solutions beyond the classical $(m, k) = (3, 1)$ are observed.

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New in this version, we include a dedicated high-scale validation script, erdos_moser_highscale_validation.py, which exhaustively extends the valuation sieve and near-miss search far beyond prior limits. This code confirms that no exceptional cases arise over millions of $(m, k)$ pairs under stringent $p$-adic constraints, substantially strengthening the empirical case for the conjecture’s correctness.

This release includes a standalone, fully-documented Python engine implementing the theoretical results and reproducing all experiments. The code is optimized for validation, extension, and discovery of further cases, offering a reusable computational toolset for future work on the Erdős–Moser conjecture and related problems in computational number theory.