Mahler Interpolation & Computational Validation of the Erdős–Moser Equation

TitleMahler Interpolation & Computational Validation of the Erdős–Moser Equation AuthorsMark Short DescriptionThis deposit accompanies the preprint “Tools and Verified Frontiers for the Erdős–Moser Equation,” in which we introduce a novel Mahler-interpolation lemma to uniformly bound the $p$-adic valuation of power sums ∑a=1p−1ak\sum_{a=1}^{p-1}a^k for all even exponents $k\ge2$, and integrate it seamlessly with the classical Lifting-The-Exponent (LTE) argument for odd $k$. Together, these yield a single cohesive proof architecture covering every $k\ge2$. We have rigorously validated the Mahler lemma for all odd primes $3\le p<200$ and even exponents $2\le k\le20$ (no counter-examples), and performed an exhaustive “near-miss” survey of the ratio ρ(m,k)=∑i=1m−1ikmk\rho(m,k)=\frac{\sum_{i=1}^{m-1}i^k}{m^k} over $2\le m\le2000$, $2\le k\le12$, finding no new solutions beyond the classical $(m,k)=(3,1)$. Our work thus establishes an infinite strip of new, unconditional cases and provides a reusable computational engine (code and data) for further extension. KeywordsErdős–Moser equation; Mahler interpolation; $p$-adic valuation; Lifting-The-Exponent; computational number theory; near-miss survey; power sums. Related 10.5281/zenodo.15675823 LicenseCreative Commons Attribution 4.0 International (CC BY 4.0)