Maybe there's something interesting about the full subcategory of FinSet on squares, Sqr? That inclusion does preserve the initial object and finite products, at least - maybe it's worth thinking about Sqr as a Lawvere theory or something? Or looking at Sets^Sqr(op), since that's the free cocompletion, and comparing the coproducts there with the coproducts in FinSet....

right, you can categorify the problem by considering, for any category C, the full subcategory C² of all “squares” in that category, ie the full subcategory on the objects {XxX | X ∈ C}…

Idk if looking at the full cocompletion will get you much? it’ll give you all the “formal sums of squares”, with an isomorphism if a formal sum is a square, I suppose…. interestingly, if a number is expressible as the sum of two squares in two different ways but isn’t itself a square, I think you’ll get two different objects in the free cocompletion.

Looking at it with a Lawvere theory perspective might be nice? idk, I don’t know much about lawvere theories, it feels like it might necessitate the creation of partial lawvere theories or something idk…

What is your opinion on the existence of a perfect cuboid (a rectangular prism with integer edge lengths, face diagonals, and space diagonal)?

Anonymous

OK, yknow what, this is a problem that’s been open forever, I’m probably not going to run out of approaches in a reasonable amount of time.

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So, here we go. The problem is to find three integers x,y,z, such that all the diagonals in the cuboid with edges of length x, y, and z, are of integer lengths.

One way to look at this is: this reduces to solving the system of four diophantine equations:

x² + y² - a² = 0
x² + z² - b² = 0
y² + z² - c² = 0
x² + y² + z² - d² = 0

Now as it happens, we know one thing about systems of quadratic diophantine equations: we know that there can be no algorithm that determines, for any such system, whether or not this system has any solutions. This is a consequence of the unsolvability of Hilbert’s tenth problem (exercise).

Thus, we know that any approach to this problem that will actually work is going to be applicable to this and possibly some other small class of diophantine problems.

So, onwards. You’ll have noticed that solving this problem in the integers and solving it in the rationals are equivalent things.

Another way of looking at this is: consider the property P that a set of (rational or natural) numbers can have:

For each nonempty subset S of this set, the sum of all the numbers in S is a square

This property obviously can’t be satisfied by infinite sets of numbers, and also it is hereditary: if it is satisfied by a set S, then it will be satisfied by any subset T of S. Therefore, the set of all sets of numbers with this property forms an abstract simplicial complex, and demonstrating the nonexistence of a perfect cuboid amounts to demonstrating that this complex is a 1-dimensional complex, i.e., a graph.

Thus, one approach would be to try and bound the dimension of this complex. I don’t know of a good way to do that though.

A third way of looking at this is: what are the squares of (rational or natural) numbers? The squares of rational numbers are a subgroup of the multiplicative group on the positive rationals, and the squares of natural numbers are a submonoid of the multiplicative monoid on the positive natural numbers.

The point being, the abstract structure of the squares of naturals is that of a monoid that distributes over a partial monoid, which is a weaker structure than a ring that you might call a bell (‘cuz get it, bells, rings…) while the abstract structure of the squares of rationals is that of a group that distributes over a partial monoid, which is sort of a field the the bell’s ring, which I’m going to call a whistle (because bells and ____ ).

Anyways, another approach would be, you could study the abstract structure of bells and whistles and deduce from it useful things maybe. At least that’s the way I’m thinking about it now, since it seems to be the approach that preserves all the algebraic structure present on the squares…

so, step 1: study the categories of bells and whistles…

to the anon who sent me an ask about what I thought about the question regarding the existence of a perfect cuboid: I am still thinking about it.

sorry guys, been kinda busy and fell behind on the Feed  _<_

anyways. new icon.