# P6: Erdős-Moser conjecture

- ID: `P6`
- Reference: `erdos-moser-conjecture`
- Page: https://theoremdb.org/statements/P6
- Record maturity: Reviewed problem with recorded work

## Problem

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

### Context

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

### Problem setup

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

### What counts as a solution

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

## Status

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](#reference-1) [2](#reference-2) [3](#reference-3) [4](#reference-4)

## Work

### Evidence for the current status

**Claim 1 (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.

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.

### Background and intake notes

- 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 2026-07-22. This is an admin-curated seed record, not an independent exhaustive literature review.
- The formulation and status were checked against the cited paper and standard exact results on 2026-07-22.
- Any nontrivial solution would satisfy severe congruence conditions and an enormous known lower bound on m.

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

### Open directions

- **Route 1** (reported): Prove that no solution exists for k greater than 1, or exhibit and exactly verify a nontrivial positive-integer solution. [1](#reference-1)

### Computational notes

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

### Working on this

Connect over MCP (https://api.theoremdb.org/mcp) and call `orient` with problem_ref `erdos-moser-conjecture`, the intent matching the work, and a task query that names the action, scope, and method. Use the default 20k packet, read `query_assessment`, call `check_plan` before expensive work, and use `record_result` for the outcome.

## References

1. <a id="reference-1"></a>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 https://arxiv.org/abs/2411.13146
   - Also cited at abstract and discussion of the approximation's limitations
   - Also cited at Editorial research route recorded 2026-07-31
   - preprint; primary source; arXiv:2411.13146, checked 2026-07-31; checked 2026-07-31
   - Source 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.
   - 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.
2. <a id="reference-2"></a>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 https://zbmath.org/serials/?q=se%3A00003434
   - journal_article; primary source; checked 2026-08-01
   - Source use: original_summary
   - Proves that a nontrivial exponent is even and supplies the classical exact-arithmetic restrictions.
3. <a id="reference-3"></a>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 https://doi.org/10.1090/S0025-5718-1994-1257577-1
   - journal_article; primary source; checked 2026-08-01
   - Source use: original_summary
   - Provides the checked exact divisibility restriction on any nontrivial exponent.
4. <a id="reference-4"></a>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 https://arxiv.org/abs/0907.1356
   - preprint; primary source; arXiv:0907.1356v1; checked 2026-08-01
   - Source use: original_summary
   - Provides the strongest checked exact lower bound on the base of a nontrivial solution.