The Erdős–Moser equation

formal-conjectures · wikipedia · ams-11

For positive integers $k$ and $m$, let

$$S_k(m)=1^k+2^k+\cdots+(m-1)^k.$$

The Erdős–Moser conjecture says that $S_k(m)=m^k$ has only the solution
$(k,m)=(1,3)$.

The only positive solution of $S_k(m)=m^k$ is $(k,m)=(1,3)$.

Mathematical status
Open: marked research open in google-deepmind/formal-conjectures at revision cd3d8db4634733a748b2380f80f77ba3e4b9dda0 (checked 2026-09-11).

Formal availability
A Prop definition is supplied for the pinned Lean 4.33.1 environment and must be accepted by the deployed verifier before it is available.

Sources and provenance
Upstream reference cited by formal-conjectures (checked 2026-09-11): https://en.wikipedia.org/wiki/Erd%C5%91s%E2%80%93Moser_equation
Upstream reference cited by formal-conjectures (checked 2026-09-11): https://arxiv.org/abs/1003.1646
Formal statement provenance (Apache-2.0) (checked 2026-09-11): https://github.com/google-deepmind/formal-conjectures/blob/cd3d8db4634733a748b2380f80f77ba3e4b9dda0/FormalConjectures/Wikipedia/ErdosMoser.lean
Local target: Corpus.WikipediaErdosMoser.erdos_moser_conjecture
Source SHA-256: 475393ece1e30ece738f2e560be7b4b06cc92669e0e43cbcab57687b4c1e441f
Lean v4.33.1; Mathlib 0df444a360eaa60ab8c11dca51a86af692955474; policy kernel-replay-v1.
The deployed accepted environment records the actual immutable verifier image.

Research discussions are not verified proofs. A formal target specifies a precise statement; accepting a target does not prove it. Checked results apply to their exact statements and pinned environments.

Public JSON record