The reciprocal sum of the prime-prefix-free numbers converges [pdf](jdb19937.github.io) |
The reciprocal sum of the prime-prefix-free numbers converges [pdf](jdb19937.github.io) |
This Lean-verified proof answers a question I posed 10 years ago: https://math.stackexchange.com/questions/2288648/does-the-su...
An equivalent version: if we start with 1 and then output a stream of random bits, reading the number as a big-endian binary number at each step (so each time a bit arrives, the number is multiplied by 2 and 1 is either added or not), the expected time until the number is an odd prime is finite.
Anyway, I think it's weird it depends on the Riemann Hypothesis.
Do you have some numerical test for intervals like sum up to 1000, up to 10000, up to 100000, up to 1000000, ... ?
?
* I totally forgot the second log in sum(1/primes) ~= log(log(N)). It's nice to see numerical experiments, but now I realize I had to agree that it's difficult to get a huge number even in the well known case that is infinite.
* The article says that the result of the version with the binary prefix is finite but version with the ternary prefix is infinite. Do you have some numerical experiments? I'd love to see a graphic with the correct amount of logs in both axes to show the difference of behaviour.
* IIUC, the result of the version with quaternary prefix is infinite too, but the result should be comparable to the result of the binary prefix. At least quaternary(N)>binary(N). [I'm not sure if ¿ternary(N)>binary(N)?. Looks difficult.] So it's totally posible (and perhaps obvious) that quaternary(N) is unbounded in spite binary(N) is bounded. It's not very intuitive, but I think I saw something very slightly similar in the past and I got surprised too.
* I'm still not sure why it uses the RH, but it looks like you really thought about it (importing lemma 2.1 and remark 7.1), so I guess I will not be able to remove the RH skimming the paper.