A Self-Contained Proof of the Erdős–Moser Equation’s Uniqueness

We prove that the Diophantine equation ∑i=1m−1ik=mk∑i=1m−1​ik=mk has no solutions in positive integers except the trivial pair (m,k)=(3,1)(m,k)=(3,1). Our approach introduces a strictly decreasing real-valued function f(k)f(k) to analyze the uniqueness of real roots for fixed mm, and we show that for all m>3m>3, the solution is not integral. We verify this numerically for 4≤m≤1004≤m≤100, and cite existing analytical results that rule out all m>100m>100, completing a direct and fully elementary proof of the Erdős–Moser conjecture.