The Erdős–Moser equation
The Erdős–Moser equation
Inferred from the community field; finer classification has not been reviewed.
Problem statement
erdos_moser_conjecture
The only positive solution of is .
Lean 4
Formal statement
Lean 4/-
Copyright 2026 The Formal Conjectures Authors.
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you may not use this file except in compliance with the License.
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-/
import FormalConjecturesUtil
/-!
# The Erdős–Moser equation
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
$(k,m)=(1,3)$.
*References:*
* [Wikipedia: Erdős–Moser equation](https://en.wikipedia.org/wiki/Erd%C5%91s%E2%80%93Moser_equation)
* B. C. Kellner,
[On stronger conjectures that imply the Erdős–Moser conjecture](https://arxiv.org/abs/1003.1646)
-/
namespace ErdosMoser
/-- The power sum $S_k(m)=\sum_{i=1}^{m-1} i^k$. -/
def powerSum (k m : ℕ) : ℕ :=
∑ i ∈ Finset.Ico 1 m, i ^ k
/-- The only positive solution of $S_k(m)=m^k$ is $(k,m)=(1,3)$. -/
@[category research open, AMS 11]
theorem erdos_moser_conjecture :
∀ k m : ℕ, 0 < k → 0 < m → powerSum k m = m ^ k → k = 1 ∧ m = 3 := by
sorry
/-- The pair $(1,3)$ is the trivial solution of the Erdős–Moser equation. -/
@[category test, AMS 11]
theorem powerSum_one_three : powerSum 1 3 = 3 ^ 1 := by
decide
/-- For $k=2$ and $m=3$, the left side is $1^2+2^2=5$, not $3^2$. -/
@[category test, AMS 11]
theorem powerSum_two_three : powerSum 2 3 = 5 ∧ powerSum 2 3 ≠ 3 ^ 2 := by
decide
/-- Zero is a solution for every positive exponent, so the conjecture must require $m>0$. -/
@[category test, AMS 11]
theorem powerSum_zero (k : ℕ) (hk : 0 < k) : powerSum k 0 = 0 ^ k := by
simp [powerSum, Nat.zero_pow hk]
end ErdosMoser
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Background
Source snapshot inspected on 10 September 2026. Subsequent research status and the accuracy of source formalizations have not been independently certified.
Fixed commit b82b08faa9006484021c12005ab41287fb2ffb69. This file contains 1 source-labeled research-open declarations. Other declarations in the attached file can be definitions, solved results or tests. The file is preserved as source material; it has not been compiled or proved by this import.
Copyright The Formal Conjectures Authors. Code: Apache-2.0; other original material: CC BY 4.0. Third-party sources retain their own licenses. Original source files and notices preserved.
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