Plenty of options f(n)= [f([n])] etc

I've just realised, I could call this a fundamental theorem of something or other. Dave - any ideas?

I don't think I understand the syntax. What do the square brackets mean?

It means you squish the function.

Very good.

cast as integer

Ah, got it! Wow, chez, how did you know that?

I knew that you were just pretending to not understand my question. But there is a function that uses primes to get a number that indicates there is a higher prime than all of the ones used. So is there a proof that nothing like that exists for the squished numbers? And if there isn't such a technique possible, is it more likely that a human or the AI you referred to will discover that?

Like this one?

Seems pretty fundamental. It's like Euclid came up with it or something.

We could always use the divergence of the harmonic series and the Euler product for the zeta function, if you like?

Sorry least greatest integer. Also known as floor/ceiling functions.

https://en.m.wikipedia.org/wiki/Floo...ling_functions

Yeah, still don’t know what you’re asking.

Has it somehow been proven that manipulating the known even numbers between twin primes will not lead you either to to a higher number that is between two primes or a number that somehow can be proven is either such an even number or is indicative that such a higher number exists.

Also is proving that using these even numbers to prove the existence of a higher one, a task that you think is easier for a machine than a human? Same question for the task of proving there is NOT a formula for the nth prime. (Notice I am not asking to find the formula if it exists. I am asking to prove there can't be one if it does not exist.)

I think DS is talking about Mersenne numbers (i.e 2^p-1) is the main way we find new candidates for extremely large primes, my completely naive recollection is that yes there is some common forms involving powers of 2 as well that are used to search for large twin primes as well.

Obviously nobody can prove that there are a finite number of "twin prime evens" and also nobody can prove that there must exist a "twin prime even" greater than any in a list of the first n for all n. Either of those would resolve the twin prime conjecture. Also, nobody can prove that an algorithm cannot exist for generating another twin prime even given some set of them.

I'm a big believer in computer intelligence and computers are getting so much better at math while humans stay the same so I'm inclined to bet that computers will play a major roll in proving something like twin primes which humans have worked on for a long time and been unable to finish. Maybe humans are more likely to solve completely bizarre like Pi in base 10 has an infinite number of 9s etc.

I haven’t thought about this in a long time, but from what I can remember generating large primes isn’t THAT hard. ~n log log n worst case and I’m guessing you can get that down with fancier methods. Of course primality testing itself is in P so you can just guess and you’ll be right 1/Log n percent of the time.

How do you know that?

I meant that as in it has not been proven. Not that it fundamentally cannot be proven.

Not sure if that answers your question.

I understood what you were saying, and I suspect David did too.

If he was asking for a citation, I don't really have one. I'm not a number theory guy, but I just haven't heard of anything remotely like proving an algorithmic lower bound on anything related to prime numbers and I assumed I would have if it had been done.

I think he was just being a bit pedantic about the wording, i.e. "nobody can prove" can mean "nobody has proved" or "it has been shown impossible to prove". I understood it to mean "nobody has proved", which is quite clearly what you meant.

I suspect if anyone does prove (or disprove) the twin prime conjecture in our lifetime, it will be Terence Tao. He is quite possibly the most talented mathematician alive and, amongst many other things, he has done a lot of research and proved some results on arithmetic progressions in the primes.

Incidentally, a string of at at least n consecutive composite positive integers (and hence a gap of at least n between 2 primes) is trivially easy to construct.

It's not pedantry at all. In maths there are things that cannot be proved and that's very differnt to things that haven't been proved.

DS probably read it in a way that is quite understandable and not at all suprising. I'd probably have read it the same when coming from a maths chap. (as opposed to a civilian). Then again he may have been questioning whather it was common knowledge that it had'nt been proved as once was the case with the fermat thingy.

Too much genius going on.

Chez, do you have a bat phone that rings whenever someone criticises David, no matter how mildly? You always seem to be on the case within 5 minutes.

I have a system of bells and flashing lights followed by electric shocks of increasing severity if I dont respond quickly enough. Trolley will tell you all about it

Obviously no denying Tao's talent, and while his most celebrated work is very adjacent to twin primes, I'm not sure someone like him, who publishes on so many diverse areas of math would be more likely than a specialist who only thinks about twin primes to prove it.

I have spent way too much time thinking about the Collatz conjecture. Alas, still no closer to proving it.