An Analytical Exploration of the Erd\"os-Moser Equation $ \sum_{i=1}^{m-1} i^k = m^k $ Using Approximation Methods

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.