
Landmark math proof clears hurdle in top Erdős conjecture - headalgorithm
https://www.quantamagazine.org/landmark-math-proof-clears-hurdle-in-top-erdos-conjecture-20200803/
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y7
The conjecture by Erdős is the following: if A ⊂ ℕ is such that Σ_{n ∈ A} 1/n
diverges, then A contains arbitrarily long arithmetic progressions. Bloom and
Sisask now proved that A contains infinitely many length-3 arithmetic
progressions, following from their main result which is an improved upper
bound for Roth's theorem.

[https://arxiv.org/abs/2007.03528](https://arxiv.org/abs/2007.03528)

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jstanley
> The conjecture by Erdős is the following: if A ⊂ ℕ is such that Σ_{n ∈ A}
> 1/n diverges, then A contains arbitrarily long arithmetic progressions

This is quite hard to understand for people who don't already know what it
means (including me). I started trying to translate it but there were a few
parts I didn't understand, starting with:

1.) Is ℕ integers >0 or >=0? Wikipedia says it can be either. Maybe it doesn't
matter? Maybe it must be >0 otherwise 1/n makes no sense?

2.) When you ask whether the sum of 1/n for all n in A diverges, how do you
know what order to sum them in? Does it diverge regardless of the order? Since
A only contains positive integers doesn't sum of 1/n for all n in A always
tend towards infinity?

EDIT: I see now that sum of positive integers doesn't always tend towards
infinity, thanks to ColinWright's comment about powers of 2 (for which the sum
of 1/n tends towards 1).

Also, since the numbers are all positive, it doesn't matter what order you sum
them in, you're basically just asking whether the sum of 1/n for all the
numbers in A is finite or not.

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maxov
Dense math can often be complex to decipher. Sometimes it feels like reading
an esoteric codebase to me!

1) Yes in this case 0 is not included in N. I’ve seen N defined both ways
depending on the context, so it can be confusing when it’s not given
explicitly.

2) By definition, a series sum is based on the limit of partial prefix sums.
E.g 1/a_1, 1/a_1+1/a_2, ... It is an interesting question mathematically, if
the sum stays the same when you arbitrarily rearrange the terms of the series.
In general the answer is no (see Riemann rearrangement theorem), but as this
series is only made up of positive reals, it can be rearranged arbitrarily
without change in how it converges (or doesn’t).

To the second part of your question, take any geometric series, i.e. A = {1,
r, r^2, ...}, then it will converge. There are other classes of series that
will converge in this case, and the conjecture is basically asking to
characterize sets with diverging series as needing to be “large and dense” in
a certain sense.

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y7
Note that a geometric series can also be a multiple, e.g. (a, a r, a r^2,
...). It converges if and only if |r| < 1.

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maxov
Good point, I was not being fully general there!

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xamuel
For people interested in arithmetic progression type theorems, one of my
favorite theorems along those lines (though technically not actually about
arithmetic progressions, but extremely closesly related), and a very under-
appreciated gem, is Hindman's Theorem.

Hindman's Theorem says that if you color the natural numbers with finitely
many colors, there must be some infinite subset D of the natural numbers such
that every finite sum of elements of D has the same color.

The proof, asontonishingly, is an application of free ultrafilters, and is
actually simple enough that someone with an advanced undergrad background in
pure math can understand it with just a few hours of reading (e.g. see [1])

[1]
[https://web.williams.edu/Mathematics/lg5/Hindman.pdf](https://web.williams.edu/Mathematics/lg5/Hindman.pdf)

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P-NP
Very promising, although the jury is still out: "The new paper is 77 pages
long, and it will take time for mathematicians to check it carefully. But many
feel optimistic that it is correct. “It really looks the way a proof of this
result should look,” said Katz, whose earlier work laid much of the groundwork
for this new result."

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dmch-1
I find the existence of number theoretic problems quite puzzling. I wonder
what are the implications about the world we can make from them.

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raincom
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.

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dmch-1
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.

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raincom
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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kevinventullo
Worth noting that the Green-Tao Theorem is a very special case of this
conjecture, using the (comparatively easier to prove) fact that Σ 1/p
diverges.

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xiaodai
Does it get us closer tot he twin prime conjecture?

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HuangYuSan
No

