31–40 of 40 posts
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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.
I think people that know 'product', 'geometric series', etc. know what distinct means. It doesn't have some special mathematical meaning here.
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 patt…
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)...
i dont get it...what is the hidden connection exactly?
The real point is that no one gets the hidden connection. There are lots of tantalizing clues in number theory that lead us to believe there is some fundamental connection between addition and subtraction but we can't state it outright. This toy problem is like a black box in that we can play with it and quantify it's behavior but we don't know (exactly) why it works.
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”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)
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.
Basically, when you take integers like {1, 2, 3, 4} or {1, 3, 5, 7} (so that there are additive relations between them) (e.g. small integers), there tends to be a lot of duplicates in the addition table. But if you take integers like {1, 2, 4, 8} or {1, 3, 9, 27} (so that there are multiplicative relations between them) (e.g. very similar prime factorizations), there tend to be a lot of duplicates in the multiplication table. In other words, it's very easy to create sets with either lots of duplicates in the addition table or lots of duplicates in the multiplication table.
The conjecture is not about which kind of set is more common (so it doesn't matter even if it's true that there are a lot “more” arithmetic progressions than geometric ones, in whatever sense), but rather says there is no way you can create a set with both sorts of collisions — it's saying that if a set has many additive coincidences it can't have many multiplicative coincidences and vice-versa.
Earlier quoted context omitted.
The real point is that no one gets the hidden connection. There are lots of tantalizing clues in number theory that lead us to believe there is some fundamental connection between addition and subtraction but we can't state it outright. This toy problem is like a black box in that we can play with it and quantify it's behavior but we don't know (exactly) why it works.
I think you mean “addition and multiplication”, not “multiplication and subtraction”... there definitely is a lot of evidence that addition and subtraction are related, but that’s just plain trivial.
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> 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
That's a good point, I'll have to stand corrected on that. Just a matter of swapping one operation for another.
The first thing to remember is that multiplication IS Addition..
Earlier quoted context omitted.
I think you mean “addition and multiplication”, not “multiplication and subtraction”... there definitely is a lot of evidence that addition and subtraction are related, but that’s just plain trivial.
Thank you, mobile commenting is a blessing and a curse.