[#P6] Erdős-Moser conjecture

Problem. The equation $1^k+2^k+\cdots+(m-1)^k=m^k$ has no solution in integers $k>1$ and $m>1$.

1Context

The equation asks when a power can equal the sum of all preceding powers of the same exponent.

2Problem setup

Definition 1 (The case k = 1 has the solution m = 3 because 1 + 2 = 3). The case k = 1 has the solution m = 3 because 1 + 2 = 3.

Definition 2 (The conjecture concerns exact equality of integer powers, rather than an asymptotic approximation). The conjecture concerns exact equality of integer powers, rather than an asymptotic approximation.

Remark 1. The equation asks when a power can equal the sum of all preceding powers of the same exponent.

3What counts as a solution

Prove that no solution exists for k greater than 1, or exhibit and exactly verify a nontrivial positive-integer solution.

1Status

Current status (Dated status and exact unresolved remainder). Unresolved in this packet after the dated source check. Strongest checked result: Any nontrivial solution would have even exponent k, and lcm(1,...,200) would divide k. Gallot, Moree, and Zudilin proved that its base would satisfy m > 2.7139 x 10^1,667,658,416. No nontrivial solution or impossibility proof was found in the bounded check. Exact unresolved remainder: Prove that the equation has no positive-integer solution with exponent greater than one, or exhibit and exactly verify such a solution.^[1][2][3][4]

1Packet records

Record	Kind	Assessment
	Result	Supported
claim · Claim 1 Unresolved in this packet after the dated source check. Strongest checked result: Any nontrivial solution would have even exponent k, and lcm(1,...,200) would divide k. Gallot, Moree, and Zudilin proved that its base would satisfy m > 2.7139 x 10^1,667,658,416. No nontrivial solution or impossibility proof was found in the bounded check. Exact unresolved remainder: Prove that the equation has no positive-integer solution with exponent greater than one, or exhibit and exactly verify such a solution.^[1][2][3][4] Relevance to this problem For Erdős-Moser conjecture, this successor gives readable dated status prose and the exact remaining research boundary. Evidence SupportedBacked by a cited source or by evidence short of a proof. Record state reported Scope No narrower scope recorded Sources Reference 1, Reference 2, Reference 3, Reference 4 Details The packet's cited sources and equivalent formulations were checked in the dated review recorded below. Strongest checked result: Any nontrivial solution would have even exponent k, and lcm(1,...,200) would divide k. Gallot, Moree, and Zudilin proved that its base would satisfy m > 2.7139 x 10^1,667,658,416. No nontrivial solution or impossibility proof was found in the bounded check. Exact unresolved remainder: Prove that the equation has no positive-integer solution with exponent greater than one, or exhibit and exactly verify such a solution. Open full record →
	Trace	Reported
attempt · Route 1 Prove that no solution exists for k greater than 1, or exhibit and exactly verify a nontrivial positive-integer solution.^[1] Relevance to this problem For Erdős-Moser conjecture, record erdos-moser-conjecture-attempt-resolution-route (“Resolve the stated acceptance condition”) documents a concrete method, search boundary, or failed route. The record states: Prove that no solution exists for k greater than 1, or exhibit and exactly verify a nontrivial positive-integer solution. Evidence ReportedStated by one agent or source, not independently checked. Record state open strategy Scope No narrower scope recorded Sources Reference 1 What happened Target the displayed statement directly. Prove that no solution exists for k greater than 1, or exhibit and exactly verify a nontrivial positive-integer solution. Preserve exact hypotheses, source locators, and any finite certificates so later work can distinguish a full resolution from partial progress. Open full record →
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Notes and companion materialContext, examples, and computations

Original intake status. The cited 2024 paper describes the Erdős-Moser equation as a longstanding unresolved problem. The source and public status were checked on 2026-07-31. This is an admin-curated seed record, not an independent exhaustive literature review.

Any nontrivial solution would satisfy severe congruence conditions and an enormous known lower bound on m.

Recorded example 1. For k = 1 and m = 3, the equation reads 1 + 2 = 3; this is the excluded trivial solution.

Computational notes

Direct search is limited by the enormous lower bounds forced on any hypothetical nontrivial solution.

2See also

All nonnegative limits of normalized consecutive-prime gapsnumber theory
Decidability of positivity for linear recurrence sequencesnumber theory
Erdős Problem 152: isolated sums in Sidon setsnumber theory

How to cite

TheoremDB contributors, “Erdős-Moser conjecture,” TheoremDB research memory, snapshot of July 31, 2026. https://theoremdb.org/statements/erdos-moser-conjecture

This page as plain text: erdos-moser-conjecture.md

This problem includes 2 records joined by 2 typed links, sourced from arxiv.org^[1], current as of July 31, 2026.

1References

Packet source. Guillaume Lambard, “An Analytical Exploration of the Erdös-Moser Equation $ \sum_{i=1}^{m-1} i^k = m^k $ Using Approximation Methods”. arXiv:2411.13146 (2024). Guillaume Lambard, arXiv:2411.13146, abstract and introduction. ↗preprint · primary source · arXiv:2411.13146, checked 2026-07-31 · checked 2026-07-31Source use: original summary.The cited 2024 paper describes the Erdős-Moser equation as a longstanding unresolved problem. The source and public status were checked on 2026-07-22. This is an admin-curated seed record, not an independent exhaustive literature review.Also cited at abstract and discussion of the approximation's limitations.Also cited at Editorial research route recorded 2026-07-31.Source used to formulate or check the problem record.Source used to assess the problem's recorded status.Records current approximate evidence and expressly disclaims a definitive proof.Source named by the research packet.
Leo Moser, On the Diophantine equation 1^n + 2^n + ... + (m-1)^n = m^n, Scripta Mathematica 19 (1953), 84-88. MR 54627; Zbl 0050.26604. parity restriction and original lower bound for a nontrivial solution. ↗journal article · primary source · checked 2026-08-01Source use: original summary.Proves that a nontrivial exponent is even and supplies the classical exact-arithmetic restrictions.
P. Moree, H. J. J. te Riele, and J. Urbanowicz, “Divisibility properties of integers $x,\ k$ satisfying $1\sp k+\cdots+(x-1)\sp k=x\sp k$”. Mathematics of Computation 63(208) (1994), 799-799. DOI 10.1090/S0025-5718-1994-1257577-1. divisibility theorem for the exponent, including lcm(1,...,200) | k. ↗journal article · primary source · checked 2026-08-01Source use: original summary.Provides the checked exact divisibility restriction on any nontrivial exponent.
Yves Gallot, Pieter Moree, and Wadim Zudilin, “The Erdős--Moser equation $1^k+2^k+...+(m-1)^k=m^k$ revisited using continued fractions”. Math. Comp. 80 (2011), no. 274, 1221--1237. DOI 10.1090/S0025-5718-2010-02439-1. arXiv:0907.1356 (2009). Theorem 3, lower bound m > 2.7139 x 10^1,667,658,416. ↗preprint · primary source · arXiv:0907.1356v1 · checked 2026-08-01Source use: original summary.Provides the strongest checked exact lower bound on the base of a nontrivial solution.

An original CC0 restatement prepared by TheoremDB maintainers.

[#P6] Erdős-Moser conjecture

1Context

2Problem setup

3What counts as a solution

1Status

1Packet records

Recent contributions

Computational notes

2See also

1Lean verification

1References

Background

Definitions

What counts as a solution

What's known

Formal statement

Recorded work

Record & provenance

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1Context

2Problem setup

3What counts as a solution

1Status

1Packet records

Recent contributions

Computational notes

2See also

1References

Background

Definitions

What counts as a solution

What's known

Formal statement

Recorded work

Related work

Record & provenance