I wrote a little program to attack this question for a smaller number of primes. The results don’t encourage me to think that there’s a “principled” answer in general. I ran it for (as it happens) the first 1002 primes, up to 7933. The visibility counts fluctuate wildly; it looks as if there may be a tendency for “typical” visibility counts to decrease, but the largest is 256 for the 943rd prime (7451) which not coincidentally has a large gap before it and smallish gaps immediately after.
It seems plausible that the winner with a billion primes might be the 981,765,348th prime, which is preceded by a gap of length 38 (the largest in the first billion primes), but I don’t know and I wouldn’t bet on it. With 1200 primes you might think the winner would be at position 1183, after the first ever gap of size 14 -- but in fact that gap is immediately followed by another of size 12, and the prime after that does better even though it can’t see its next-but-one neighbour, and both are handily beaten by the 943rd prime which sees lots of others above as well as below.
It’s still feeling to me as if any solution to this is going to involve more brute force than insight. Thomas, would you like to tell us whether you know of a solution that doesn’t involve a lot of calculation? (Since presumably any solution will at least need to know something about all the first billion primes, maybe I need to be less vague. If the solution looked like “the winning prime is the prime p_i for which p_{i+1}-p_{i-1} is greatest” or something similarly simple, I would not consider it to be mostly brute force.)
Well, congratulations for what you have done so far.
I have hoped it will be something like this. An intricated landscape of prime towers. I don’t have a solution yet because I have invented this problem this Monday morning. Like “Oh, My God, it’s Monday morning, I have to publish another Problem on my blog and cross-post it on Lesswrong …”.
I did some Googling to prevent my brains to plagiarize too much, and that was all.
I doubt, that there is a clever solution, just some brute force solutions are possible here. But one has to be clever to perform a brute force solution in this case.
I wrote a little program to attack this question for a smaller number of primes. The results don’t encourage me to think that there’s a “principled” answer in general. I ran it for (as it happens) the first 1002 primes, up to 7933. The visibility counts fluctuate wildly; it looks as if there may be a tendency for “typical” visibility counts to decrease, but the largest is 256 for the 943rd prime (7451) which not coincidentally has a large gap before it and smallish gaps immediately after.
It seems plausible that the winner with a billion primes might be the 981,765,348th prime, which is preceded by a gap of length 38 (the largest in the first billion primes), but I don’t know and I wouldn’t bet on it. With 1200 primes you might think the winner would be at position 1183, after the first ever gap of size 14 -- but in fact that gap is immediately followed by another of size 12, and the prime after that does better even though it can’t see its next-but-one neighbour, and both are handily beaten by the 943rd prime which sees lots of others above as well as below.
It’s still feeling to me as if any solution to this is going to involve more brute force than insight. Thomas, would you like to tell us whether you know of a solution that doesn’t involve a lot of calculation? (Since presumably any solution will at least need to know something about all the first billion primes, maybe I need to be less vague. If the solution looked like “the winning prime is the prime p_i for which p_{i+1}-p_{i-1} is greatest” or something similarly simple, I would not consider it to be mostly brute force.)
But then again. Now, I think that there is a non-brute-force solution.
Well, congratulations for what you have done so far.
I have hoped it will be something like this. An intricated landscape of prime towers. I don’t have a solution yet because I have invented this problem this Monday morning. Like “Oh, My God, it’s Monday morning, I have to publish another Problem on my blog and cross-post it on Lesswrong …”.
I did some Googling to prevent my brains to plagiarize too much, and that was all.
I doubt, that there is a clever solution, just some brute force solutions are possible here. But one has to be clever to perform a brute force solution in this case.
Which you guys are.