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 problem in number theory, with the only known solution in positive integers being $ (k, m) = (1, 3) $. This paper investigates the possibility of other solutions by employing approximation methods based on the Euler-MacLaurin formula to extend the discrete sum $ S(m - 1, k) $ to a continuous function $ S_{\mathbb{R}}(m - 1, k) $. Analyzing the approximate polynomial $ P_{\mathbb{R}}(m) = S_{\mathbb{R}}(m - 1, k) - m^k $, we apply the rational root theorem to search for potential integer solutions. Our investigation confirms that for $ k = 1 $, the only solution is $ m = 3 $. For $ k \geq 2 $, the approximation suggests that no additional positive integer solutions exist. However, we acknowledge the limitations of using approximation methods in the context of Diophantine equations, where exactness is crucial. The omission of correction terms in the approximation may overlook valid solutions. Despite these limitations, our work provides insights into the behavior of the Erd\"{o}s-Moser equation and highlights the challenges in finding solutions using analytical methods. We discuss the implications of our findings and suggest directions for future research, emphasizing the need for exact analytical techniques to conclusively address the conjecture.