@Conflux Also missing “positive” here

Nevermind, I was misinterpreting “distinct perfect square integers” as “perfect squares of distinct integers

Market without the time limit:

What is with this volatility?? Throwing in a bit of subsidy

@Conflux I think the probability of proving this is very small (unless we have AGI by then). Basically, these kind of problems are usually very hard to prove, I don't think that many people work on it and also the base rate of a concrete theorem being proven in 2.5 years is just very small. Although there is a new result in this direction, as far as I know it is only up to constantly many exceptions. I do not know their technique but, again, a priori it's not clear how to push techniques that prove stuff "up to finitely many exceptions" to prove that in fact there are none.

This paper shows that the set of real 3x3 magic squares are parametrized by x,y,z and are of the form (this is what the "equations" of the video reduce to)

[[x+y, x+z, x-y-z],[x-2y-z, x, x+2y+z],[x+y+z, x-z, x-y]]

Since x occurs in the middle, x must be an integer, and from the upper left and upper center, y and z must likewise be integers, so all magic squares of integers are exactly squares of this form for x,y,z integers.

Letting w = y+z, we can rewrite the entries as

[[x+y, x+w-y, x-w],[x-w-y, x, x+w+y],[x+w, x-w+y, x-y]]

So finding a magic square of squares amounts to finding x,w,y such that x + n w + m y is a square for n,m in {-1,0,1}. This makes several sets of arithmetic progressions of length 3 among the values.

This is all presumably still subject to the math he was talking about, but I think it clarifies the problem somewhat.