Problem dossier · Number Theory
Odd perfect numbers
Does an odd number exist that equals the sum of its proper divisors? Every known perfect number is even.
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Status
Open for over two millennia. Any odd perfect number exceeds 10^1500 and must satisfy a long list of congruence and factor conditions (Euler form q^k m^2).
Think you can crack this one? Read the playbook before you announce →
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The Angle of Attack
Tighten the web of constraints computationally; search for spoofs (Descartes numbers) whose structure shows exactly which conditions bite; push factor-chain arguments in feasible ranges.
Tags: perfect numbers · divisors · sigma function · bounds
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The Lab
No instruments built yet. When this problem gets tackled, its interactive instruments — explorers, searches, verifiers running in the browser — live here. See the Collatz dossier for what a fully tackled problem looks like.
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The Log
Empty. Work on this problem gets logged here as dated entries — constructions tried, code run, dead ends included. Dead ends are results.
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