If you’re not familiar with the problem I’m semipublicly trying to solve, please start reading here. Coming to think about it, this is basically just another polymath-project. With the only difference that not a lot of people participate 😛 Hopefully that’ll change though. Anyway, the heuristics and assumptions in the previous post may seem a bit weird, so I try to make things more senseful here; let be given. Assume with and where the run over all primes such that . Then if, and only if does not divide for all . Let be the amount of for which and (so is the function we want an decent upper bound on). Then the ‘chance’ that doesn’t divide (assuming for simplicity that for all , ), equals . The reasonable assumption (and possibly not even that hard to make rigorous) seems that and are independent for . So the ‘chance’ that does not divide for all , should be . And if we have a good upper bound on , say , then we should be able to show that this product doesn’t go to fast enough to prevent from happening.
More sensible heuristics