In Praise of Stupid Questions
mathenchant.wordpress.com
mathenchant.wordpress.com
I have an extremely vague question; Is there one of these "stupid" ways of computing pi that doesn't involve an infeasible (to humans) infinity? I'm comparing to the "have pick people random integers and the probability they are coprime is 6/pi^2" method, which, again, to really work involves some poor people wasting an infinity of their lives. Your scheme does too from what I understand? Is this necessary?
Off topic: If you search for "quantum bernoulli factory" you will find some work I did that shows f(p)=2p is achievable if your "quantum coins" are presented as coherent superpositions instead of classical incoherent mixtures. Your work on exact sampling completely blew my mind (I'm a physicist!) while I was trying to undersantd that whole field.
My first thought does not achieve the task: expand pi/4 as a sum of positive rational numbers, then have each human use a couple of fair coins from their own Bernoulli factory to output a coin that is heads with probability given by their assigned rational number. The n'th human gets told the partial sum up to that point, flips their bernoulli factory coin and either terminates the protocol if they get heads, otherwise they adds their term to the partial sum they received and passes it on. The problem is the information content in the partial sums will grow with n.
Give the humans an order 1,2,3,…, and let a referee read them in that order. Person k tosses one fair coin and reports H/T. The referee stops at the first time the reported heads exceed the reported tails. If N people were consulted, choose one of those N uniformly at random. Output heads iff that chosen person’s coin was heads.
For a one-pass version: instead of storing the whole consulted prefix, the referee can keep a single “currently marked” consulted person, and when the k-th consulted coin arrives, replace the mark by that new person with probability 1/k. When the process stops, the marked person is uniform among the consulted ones.
"I suppose that, on a practical level, a take-home for the practicing mathematician is that if you use ChatGPT, don’t trust it to generate valid proofs, and even when it finds a valid proof, don’t be so sure it’s a good proof. And whatever you do, don’t have ChatGPT create a bibliography for you."
In isolation, that's all very good advice. But however is a take-home from what goes before? You used ChatGPT, it did generate a valid proof, it was "a solid by-the-book argument that employed a method I’ve used myself" which may or may not imply "a good proof" but suggests it was at least OK, and nothing in the story you told involves ChatGPT generating bibliographies at all.