One great problem, three kinds of immersion
From a story to your own conjecture
Problem stories open curiosity. The problem workshop is the next place, where curiosity becomes a testable idea.
- 1 · DiscoverMeet the questionLearn when it appeared and why it still holds people’s attention.
- 2 · ChallengeTest one caseMake your first observation in five minutes with a drawing, calculation, or colors.
- 3 · DevelopYou are hereBuild on ideasPublish an observation and grow it through comparisons, conjectures, and counterexamples.
Odd Perfect Number — Does One Exist?
Problem
A perfect number equals the sum of its proper divisors: Every known perfect number is even. Does an odd perfect number exist?
Why it matters
Euclid (~300 BCE) gave a construction of even perfect numbers from Mersenne primes, and Euler later proved all even perfect numbers have that form. The explicit existence question appears in Descartes's 1638 letter to Mersenne. In the 388 years since, not one odd example has been found and no impossibility proof is known.
Progress so far
No odd perfect number exists below (Ochem–Rao 2012). If one exists, it must have at least 101 prime factors counted with multiplicity, at least 9 distinct prime factors, and a largest prime factor above . The constraints are severe but still not contradictory.
Further reading
💡 Explore together, one line at a time(0 contributions)
Contributions are not ranked by popularity. Curator feedback names what is clear or reproducible, and peer signals mean someone understood or actually reproduced it.
What did you notice?
You do not need a complete proof. A small observation can open the next path.
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