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dbaupp comments on Where Fermi Fails: What is hard to estimate? - Less Wrong
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The 4-millionth (4000000th) prime number. As soon as I typed it I wondered if the distribution of large prime numbers had been approximated. I found http://en.wikipedia.org/wiki/Prime_number_theorem
It seems more than plausible that Feynman would have been familiar with that theorem and would have been able to get the 4000000th prime number from it, within 10%. If he would have, I would have been happy with my failure to stump him.
Without looking it up (or clicking that link): 54,000,000?
(Vaguely remembering that it's about
n log n(maybe?), and attempting to approximate log in my head.)(EDIT: I was almost within 20%... but even that approximation above isn't within 10%.)
n log n calculated correctly is 11% off, so it almost seems like a deliberate trick! I certainly did not know the answer when I set it and thought n log(n) was right, but then looked it up in wik and thought I was confusing it with Stirling's approximation for large factorials.
When I tried n log(n) in my head, I was off by a factor of 10. I realized after the fact is because I did it as log10(4e6) = 66 dB and then 66log(10) = log(4e6). But of course dB are 10log10() and I forgot to divide by 10 at the end.
So I'm guessing Feynman would have gotten within 12% using n log n, and then knowing Feynman, he would have known that n log n underestimates primes and since he was aiming for 10% error he would have taken 10% off his answer and nailed it.
Screw AI, let's just build Feynman when we get the technology. He was a hoot!
Sounds good to me.