Erdős–Moser problem

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

For integers $m\ge2$ and $k\ge1$, the Erdős--Moser equation is \[ \sum_{j=1}^{m-1}j^k=m^k. \] The pair $(m,k)=(3,1)$ gives $1+2=3$.

Statement recorded in the supplied source.

Is $(m,k)=(3,1)$ its only solution with $m\ge2$ and $k\ge1$? Equivalently, does it have no solution with $k\ge2$? [S2]

Short English statement

Is one plus two equals three the only positive-integer power-sum equality of this form?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 9400105 (AMR-093-0105). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Pieter Moree, Moser's mathemagical work on the equation $1^k+\cdots+(m-1)^k=m^k$, arXiv:1011.2940v2 (2011) Introduction, equation (1), p. 1. https://arxiv.org/pdf/1011.2940.
• S3 Wikipedia contributors, List of unsolved problems in mathematics, revision 1366636767 (2026) Number theory, Erdős–Moser problem. https://en.wikipedia.org/w/index.php?title=List_of_unsolved_problems_in_mathematics&oldid=1366636767#Number_theory.