autistic

surprised at people doing weird shit

?????

I mean, the actual answer is severalfold: "sometimes, when you need to fill a space, you don't end up with simple compound numbers of identical packages" is one, but really, it's a problem in mathematics which, were we to have a general solution to find the most efficient method of packing n objects with identical properties into the smallest area, we would be able to more effectively predict natural structures, including predicting things like protein folding, which is a huge area of medical research. Simple, seemingly inapplicable cases can often be generalised to more specific cases, and that's how you get the entire field of applied math, as well as most of scientific and engineering modeling

Mathematicians try this with every number

Basically just to see if they can. We can think of the problem from multiple angles. The general problem is: "if we have a larger square with side length of a, what's the maximum number of smaller squares (with side length of b) that we can fit into that larger square?". If we have a larger square with side length of 4, then we can fit 16 squares in. If the larger square had a side length of 5, then we can fit 25 squares in. So this means that if we want a neat packing solution, and we can control how large the outer square is (in relation to the inner squares), then we want each side of the larger square to be a whole number multiple of the smaller square's side length.

But what if that isn't our goal? The fact that packing 25 squares into a 5x5 square is an optimal packing solution with no spare space means that it will be impossible to fit 25 smaller squares into a square that's less than 5x5 large. But what about if we do have awkward constraints, and the number of smaller squares we have to pack isn't a square number? The fact that this weird packing solution in the OP has 17 squares isn't because 17 is prime, but rather that 17 is 1 more than 16 (it's just that 17 happens to be prime).

This is a long way of saying that because packing 16 squares into a square is easy, the natural next question is "how large does the larger square need to be to be able to pack 17 squares into it?". If this were a problem in real life where I had to pack 17 squares into a physical box, most people would just get a box that's at least 5x5 large, and put extra packing material into all the spare space. But asking this question in terms of "what's the smallest possible box we could use to pack 17 squares in?" is basically just an interesting puzzle, precisely because it's a bit nonsensical to try to pack 17 squares into the larger square. We know for certain we need a box that's larger than 4x4, and we also know that we can do it in a 5x5 box (with a heckton of spare space), so that gives us an upper and lower bound for the problem — but what's the smallest we could use, hypothetically?

As a fellow autistic person, I relate to your confusion. But I'd actually wager that there were a non-zero number of autistic people who were involved in this research. It sort of feels like "extreme sports" for autistic people — doing something that's objectively baffling, precisely because it feels so unnatural and wrong

It's not just primes.

https://en.wikipedia.org/wiki/Square_packing