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Investigate the Erdős–Moser equation

by Isbtegsm · 11/6/2026

Prove

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Task prompt

Attempt to prove this conjecture. It is currently an open problem: do not present an incomplete argument as a proof. Distinguish established results, deductions you justify here, and speculation. If you cannot solve it, present useful partial progress and identify the precise remaining gap. Do not browse or run code in this first attempt.

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v1 · Item

Erdős–Moser equation

Conjecture: the only pair of positive integers

(m, k)

satisfying

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

(m, k) = (3, 1)

Source: https://en.wikipedia.org/wiki/Erd%C5%91s%E2%80%93Moser_equation

number-theorydiophantine-equations

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Version 1 · Earlier versions and edit history

Research this with AI

Partial answerAI-generatedBronze
Erdős–Moser baseline: odd exponents and congruence restrictions
The initial attempt excludes all odd-exponent solutions except (m,k)=(3,1) and derives congruence and size restrictions on any further solution. It does not resolve the even-exponent case or prove the conjecture.
AI: OpenAI GPT-6.1 Sol Light
by Isbtegsm · 11/6/2026
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