
A Strange Grid Reveals Hidden Connections Between Simple Numbers - headalgorithm
https://www.quantamagazine.org/the-sum-product-problem-shows-how-addition-and-multiplication-constrain-each-other-20190206/
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avip
Overview of the sum/product problem by Tao
[https://terrytao.wordpress.com/2007/12/06/milliman-
lecture-i...](https://terrytao.wordpress.com/2007/12/06/milliman-lecture-iii-
sum-product-estimates-expanders-and-exponential-sums/)

[e: that lecture is almost a decade older than the result mentioned in OP]

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qubex
I studied applied mathematics but I have to admit number theory has always
been, and probably always will be, a total mystery to me. My whole intuition
is tied to dynamics and computability, and this is just absolutely alien. I’m
probably more in awe of number theorists than the average person because
whereas they group them with mysterious ‘mathematicians’ (with all their
attendant mystique and implied genius) I’ve actually “been there, [tried] to
do that” and failed, and that marks it out as an almost insurmountable peak.

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desc
My intuition was already screaming 'it'll be the other way around for a
geometric progression' before I read that far, but I'm damned if I can
understand or even speculate why. Most likely I don't properly understand it.

I'm inclined to wonder if there's a third operator which could be tested like
this, such as exponentiation, but that's not commutative over integers.

Of course, if there are similarly intriguing patterns for noncommutative
operators (and their sequences) the obvious next step would be to look at
complex numbers and quaternions...

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thaumasiotes
> My intuition was already screaming 'it'll be the other way around for a
> geometric progression' before I read that far, but I'm damned if I can
> understand or even speculate why.

It's a question of forcing the number of sums/products to be low. An
arithmetic progression forces a large number of identical sums for the obvious
reason: (a+b) = ((a-k)+(b+k)) = ((a-2k) + (b+2k)), and so on, and those
differences of _k_ are... the definition of an arithmetic progression.

The reason a geometric progression produces a lot of identical products is
exactly the same. (ab) = (a/k · bk) = (a/kk · bkk)...

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CamperBob2
Maybe I don't understand the full implications of the problem, but it seems
obvious that a multiplicative table will result in more 'numerical diversity'
than a summation table. Both multiplication and addition yield points on lines
described by the usual mx+b equation. Addition constrains x to 1, so all of
the results will be clustered near the origin, with more coincidences as an
unavoidable result. Conversely, if x is allowed to exceed 1, the resulting
'lines' will spread out on the graph and leave more room for unique points.

What am I missing?

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Someone
_”but it seems obvious that a multiplicative table will result in more
'numerical diversity' than a summation table”_

That isn’t true. If you pick the set of numbers _{1, 2, 4, 8}_ , you get 10
different sums (2, 3, 4, 5, 6, 8, 9, 10, 12, and 16) but only 7 different
products (1, 2, 4, 8, 16, 32, and 64)

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CamperBob2
Sure, you can always find special-case exceptions, but I still don't see why
the result is surprising in the general case. There are a lot more arithmetic
progressions than geometric ones.

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AnimalMuppet
> There are a lot more arithmetic progressions than geometric ones.

I'm pretty sure that's not true. An arithmetic progression is x[n] = x[n-1] +
k and a geometric progression is x[n] = x[n-1] * k

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CamperBob2
That's a good point, I'll have to stand corrected on that. Just a matter of
swapping one operation for another.

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alanbernstein
If you consider all finite sequences of positive integers with all values
below some limit M, it might be the case that there are more arithmetic
sequences than geometric sequences. I don't think that's relevant to the
article, but might be where your head was when you made that post.

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csours
It's almost like one dimension of entropy has been exhausted.

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basicplus2
"A long-standing puzzle seems to constrain how addition and multiplication
relate to each other."

The first thing to remember is that multiplication IS Addition..

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IshKebab
Good job they explained what "distinct" means...

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vanderZwan
This is maths we are talking about, you should never assume that the common
usage of words applies to the precise definition used in the maths context.
Take the Boolean inclusive "or", for example.

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IshKebab
I think people that know 'product', 'geometric series', etc. know what
distinct means. It doesn't have some special mathematical meaning here.

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vanderZwan
Sure, but this is not an article aimed at those people. Note that geometric
progression is also defined later on.

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sunstone
This kind of feels like it's touching on some of the same themes as the ABC
conjecture.

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xiaoxiae
I have been tree and

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pwaai
i dont get it...what is the hidden connection exactly?

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Dylan16807
'connection' is a somewhat misleading way of stating it. It's more like the
opposite of a connection. A set of numbers cannot have trait A and trait B at
the same time.

Trait A is that the numbers have a certain kind of similarity in their
spacing, leading to few unique sums. Trait B is that they have a certain kind
of similarity in their factors, leading to few unique products.

So from some angle it's interesting that you can't impose both types of
patterns on a single set of numbers, because that inability implies a relation
between two very different operations.

But on the other hand why would you assume that you _should_ find an
intersection between two restrictive algorithms? Maybe there's a hidden
assumption that the algorithms would be random-ish and let you find an overlap
if you search hard enough, and that assumption is screwing with people's
intuition.

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function_seven
> _Trait A is that the numbers have a certain kind of similarity in their
> spacing, leading to few unique sums. Trait B is that they have a certain
> kind of similarity in their factors, leading to few unique products._

This is an excellent way to describe it. I had the sense that “of course the
geometric sequences will collide more when multiplying” but if you asked me
“why?” I would have trouble explaining. So thanks.

As far as finding a set that has an equal number of distinct sums and
products, would it be possible to come up with a sequence that’s
“half”-arithmetic and “half”-geometric?

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Dylan16807
Even a single value that doesn't fit in one sequence will cause a lot of extra
values, so your ability to fit multiple patterns gets worse and worse as you
go longer. Something like 2 4 6 8 16 does pretty well for its size but even
then it's giving you twelve distinct outputs for either operation.

