- I don't really know anything about Fermat numbers, but my shoot-from-the-hip feeling would be that knowing something is a Fermat number doesn't help you much more than knowing it's a Mersenne number (though of course they grow faster). So I suspect your problem has the nasty "quasi-random" character that makes it intractable.
But then I would have said that for Goldbach's conjecture too - every odd number ≥7 is the sum of 3 primes. Some people are insanely good at finding patterns amid quasi-randomness! First Hardy and Littlewood showed this conjecture is true for sufficiently large odd numbers if the generalized Riemann hypothesis holds. Then someone showed that ≥ 3^3^15 counts as sufficiently large. Then someone showed the conjecture really does follow from the generalized Riemann hypothesis... using a computer search to handle certain numbers 10^20. Then Harald Helfgott claimed to prove it straight out. His proof was accepted for publication, but he's been working on releasing it ever since then:
https://webusers.imj-prg.fr/~harald.helfgott/anglais/book.html
All of this is completely beyond my ken.
47% of people on the prediction market think a Millennium problem will be solved by AI before 2028. I'd like to ask them which one is the most likely.
https://en.wikipedia.org/wiki/Goldbach%27s_weak_conjecture
John Carlos Baez
npub1nf…3nqe4
2026-08-16 15:33:47 UTC
in reply to nevent1q…nwnz
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2026-08-16 15:33:47 UTCEvent JSON
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"content": "- I don't really know anything about Fermat numbers, but my shoot-from-the-hip feeling would be that knowing something is a Fermat number doesn't help you much more than knowing it's a Mersenne number (though of course they grow faster). So I suspect your problem has the nasty \"quasi-random\" character that makes it intractable. \n\nBut then I would have said that for Goldbach's conjecture too - every odd number ≥7 is the sum of 3 primes. Some people are insanely good at finding patterns amid quasi-randomness! First Hardy and Littlewood showed this conjecture is true for sufficiently large odd numbers if the generalized Riemann hypothesis holds. Then someone showed that ≥ 3^3^15 counts as sufficiently large. Then someone showed the conjecture really does follow from the generalized Riemann hypothesis... using a computer search to handle certain numbers 10^20. Then Harald Helfgott claimed to prove it straight out. His proof was accepted for publication, but he's been working on releasing it ever since then:\n\nhttps://webusers.imj-prg.fr/~harald.helfgott/anglais/book.html\n\nAll of this is completely beyond my ken.\n\n47% of people on the prediction market think a Millennium problem will be solved by AI before 2028. I'd like to ask them which one is the most likely. \n\nhttps://en.wikipedia.org/wiki/Goldbach%27s_weak_conjecture",
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