The Erdős–Moser Equation

Question: solve \(1^{k}\) + \(2^{k}\) + ⋯ + (k−1)^k = \(k^{k}\) in integers k ≥ 2. Moser proved k > 10^(\(10^{6}\)) for any solution; Moree and co-authors later pushed related congruence constraints further.

A deceptively innocent Diophantine equation circulating since the 1950s. The elementary observation that the left side is ≡ Σ \(i^{k}\) mod k, combined with von Staudt–Clausen structure of Bernoulli denominators, forces k to be enormous — Moser's celebrated double-exponential bound. Every refinement since has only raised the floor. Whether the answer is 'no solutions' or 'one monstrous example' is genuinely open, making this the purest tiny-question-big-mathematics specimen on the shelf.