The Erdős–Moser equation

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

From the catalogue. Imported from The Formal Conjectures Authors (Google DeepMind and contributors) (Apache-2.0) — original. Nobody has started on it here yet: tasks are created as soon as someone asks for one or submits a claim. A Lean proof is checked against the statement below by the Lean kernel; a curator confirms before the problem counts as resolved.

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Cite

@misc{cairn-erdos-moser,
  title        = {The Erdős–Moser equation},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-moser}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-28}
}

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The problem

The question

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

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

$S_{k} (m) = 1^{k} + 2^{k} + \dots + (m - 1)^{k} .$

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

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.Wikipedia.ErdosMoser.

theorem erdos_moser_conjecture :
    ∀ k m : ℕ, 0 < k → 0 < m → powerSum k m = m ^ k → k = 1 ∧ m = 3

What counts as progress

A Lean proof of the pinned statement (or of its negation, for a yes/no question) — checked by the Lean kernel against the upstream statement; a curator confirms before the problem is marked resolved.
Partial results: special cases, weaker bounds, reductions — as verified claims.
Computations and numerical evidence with published code (reproducible).
Literature: the problem may have been solved or partly solved already. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

Source and licence

Imported from Formal Conjectures (Wikipedia), commit e6d1743831c2. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.