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Landmark math proof clears hurdle in top Erdős conjecture

quantamagazine.org

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Re: Landmark math proof clears hurdle in top Erdős conjecture

#41

Post by jchook »

peterlk wrote: post #32
ColinWright wrote: post #27

Earlier quoted context omitted.

OK, let's have a go. We are looking for set of three numbers that are "equally spaced". So {4, 7, 10} are equally spaced, differing by 3 each time. Another set might be {20, 30, 40}, this time differing by 10. We'll call such a set "Equally Spaced Triples", or "EST" for short. If you have the positive even integers - 2, 4, 6, 8, ... - then clearly you can find infinitely many ESTs. You have {2,4,6}, {6,10,14}, and so…

Thank you very much for this comment I would like you to know that I would gladly pay a monthly fee to have at least 1 proof (per month) explained to me like this. Not sure how that could scale to varying knowledge levels, but I would love to have a slow educational drip of math explanations.

I second this!

Not math-focused, but you might love The Morning Paper:

https://blog.acolyer.org/

Re: Landmark math proof clears hurdle in top Erdős conjecture

#42

Post by hansvm »

hansvm wrote: post #40
Ar-Curunir wrote: post #15

Earlier quoted context omitted.

Hmm is there an asymptotic statement underlying this? The powers of 2 are exponentially sparse (N integers contain at most 1/logN powers of 2), whereas primes are polynomially sparse (N integers contains N/LogN primes)

There can be. Let f(n) be the count of members of the set in question which are at most n. If ε>0 and f(n)=O(n^(1-ε)) then the sum of reciprocals converges. For denser sets I'd have to think a bit longer about it. It's probably also worth noting that weird density distributions are possible. E.g. one could imagine a kind of oscillation where you include enough values to get the total density up to that point up to so…

Edit: not greater than exponentially sparse sets, something a bit denser -- exponential sparsity is manageable as well, but not with the 2^-k construction I gave.

Re: Landmark math proof clears hurdle in top Erdős conjecture

#43

Post by ColinWright »

gigatexal wrote: post #31
submeta wrote: post #28

Earlier quoted context omitted.

That is a brilliant explanation for non-mathematicians. Thank you very much.

Yes agreed! Thank you!!! Also updated my profile

Quick comment ... I have no idea about the answer to your email riddle. I don't know anything about Basecamp, Apple, WWDC, or any ruckus. So if you want to obfuscate your email address and yet still be contactable by humans who use computers, you need not to rely on specific knowledge from that world.

Of course, a perfectly reasonable stance is that if someone really wants to contact you then they'll do their homework and solve the riddle, in which case it's fine. I just thought I'd let you know that not everyone lives in the world where that kind of knowledge is commonplace.

Re: Landmark math proof clears hurdle in top Erdős conjecture

#44

Post by gigatexal »

ColinWright wrote: post #43
gigatexal wrote: post #31

Earlier quoted context omitted.

Yes agreed! Thank you!!! Also updated my profile

Quick comment ... I have no idea about the answer to your email riddle. I don't know anything about Basecamp, Apple, WWDC, or any ruckus. So if you want to obfuscate your email address and yet still be contactable by humans who use computers, you need not to rely on specific knowledge from that world. Of course, a perfectly reasonable stance is that if someone really wants to contact you then they'll do their homewor…

Ahh thanks for that. I can get rather deep in my Apple obsessions let me fix it.

Updated. I hope that’s better. I had just saw your profile and saw all the spam you got and thought I had to come up With a really obscure way to obscure it. Sorry.

Re: Landmark math proof clears hurdle in top Erdős conjecture

#45

Post by dmch-1 »

raincom wrote: post #17
dmch-1 wrote: post #11

I find the existence of number theoretic problems quite puzzling. I wonder what are the implications about the world we can make from them.

Aha, it’s an ontological question. Most mathematicians believe that numbers, sets, etc don’t exist in the world we live in, but exist in a special world called “Platonic world”. That’s what Platonism in philosophy of mathematics is about.

I agree that numbers don't exist in the real world. However, we humans have an ability to perceive the world through numbers. The question then is about our perception of the world if you wish.

Re: Landmark math proof clears hurdle in top Erdős conjecture

#46

Post by raincom »

dmch-1 wrote: post #45
raincom wrote: post #17

Earlier quoted context omitted.

Aha, it’s an ontological question. Most mathematicians believe that numbers, sets, etc don’t exist in the world we live in, but exist in a special world called “Platonic world”. That’s what Platonism in philosophy of mathematics is about.

I agree that numbers don't exist in the real world. However, we humans have an ability to perceive the world through numbers. The question then is about our perception of the world if you wish.

There is a subtle issue lurking behind. And that can be framed as a question: can one access things(say, numbers) that do not exist? If the answer is "Yes", then we don't need the Platonic world. Otherwise, we need to postulate the existence of numbers and sets in another world (Platonic world). You can also see people who are looking for Platonic love.

In other words, it is an issue between access and existence. Does access need existence? For instance, when X says 'John is charismatic', is 'charisma' like 'neurosis'? Definitely not, and X sees John's charisma. And this charisma doesn't play the causal role.

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