An Analytical Exploration of the Erdös-Moser Equation $ \sum_{i=1}^{m-1} i^k = m^k $ Using Approximation Methods (2411.13146v3)

Abstract: The Erd\"{o}s-Moser equation $ \sum_{i=1}^{m - 1} i^k = m^k $ is a longstanding challenge in number theory, with the only known integer solution being $ (k,m) = (1,3) $. Here, we investigate whether other solutions might exist by using the Euler-MacLaurin formula to approximate the discrete sum $ S(m-1,k) $ with a continuous function $ S_{\mathbb{R}}(m-1,k) $. We then analyze the resulting approximate polynomial $ P_{\mathbb{R}}(m) = S_{\mathbb{R}}(m-1,k) - m^k $ under the rational root theorem to look for integer roots. Our approximation confirms that for $ k=1 $, the only solution is $ m=3 $, and for $ k \geq 2 $ it suggests there are no further positive integer solutions. However, because Diophantine problems demand exactness, any omission of correction terms in the Euler-MacLaurin formula could mask genuine solutions. Thus, while our method offers valuable insights into the behavior of the Erd\"{o}s-Moser equation and illustrates the analytical challenges involved, it does not constitute a definitive proof. We discuss the implications of these findings and emphasize that fully rigorous approaches, potentially incorporating prime-power constraints, are needed to conclusively resolve the conjecture.