## Background

The following question was first asked by Alex Rice, who was thinking about small subsets $A\subset [1,\ldots , N]$ with lots of square differences. Certainly for any set $A$ the maximum number of square differences is going to be $\binom{|A|}{2}$. From the point of view of someone working in additive combinatorics, an infinite set of positive integers can't get much less substantial than the squares, and so it's natural to wonder if there are arbitrarily large sets $A$ inside the squares, all of whose differences are squares [edit: I apparently misunderstood the original motivation, see Alex's answer/comment below]. This question was asked of a few others, including Adrian Brunyate, Jacob Hicks and Nathan Walters before it was asked of me by Adrian in this form:

Definition:We say that a sequence $(a_1, \ldots, a_n) \in \mathbf{Z}^n_{\ge 1}$ is aSuper-$n$if for all $1 \le i \le n$, $a_i$ is an integer square and for all $1 \le i < j \le n$, $a_j - a_i > 0$ is also an integer square.

Clearly a Super-2 defines a Pythagorean triple.

Perhaps less clearly, a Super-3 defines an Euler Brick, and is strongly related the the question of whether there is a perfect rational cuboid.

Question 1:For which positive integers $n$ does there exist a Super-$n$ ?

If the answer is yes to the above question, we may also ask the following:

Question 2:For which positive integers $n$ do there exist infinitely many Super-$n$'s?

One may note that the following problems are related to some problems already asked on MO about rational polytopes and sequences of squares

## What seems to be known already

It has been known for millenia that there are infinitely many Pythagorean Triples.

Euler discovered in 1772 that there are infinitely many Super-$3$'s, and in fact he gave a parametrized family of them.

None of us have been able to find a Super 4 (although I haven't been searching myself).

## The connection to algebraic geometry

Definition:TheSuper-$n$-varietyis the intersection of the following $\binom{n}{2}$ quadratic polynomials in projective space over $\mathbf{Q}$. $$d_1^2 = c_2^2 - c_1^2$$ $$\vdots$$ $$d_{\binom{n}{2}}^2 = c_{n}^2 - c_{n-1}^2$$

Clearly the Super-2 variety is a copy of $\mathbb{P}^1_{\mathbf{Q}}$.

In Section 8 of the link given above for Euler's family of "Euler Bricks" we see that the Super-3 variety is birational to a singular K3 surface of Mordell-Weil rank 2. In this setting, one could say that Euler found a rational curve on this variety. It is also noted in the article that Narumiya and Shiga found a different rational curve on this variety.

Question 2':Could there be rational curves on the Super-$n$ variety for all $n$?

But perhaps (probably) this is way too much to ask. More generally, I'd like to know:

Question 3:Is there any interesting geometry to the Super-$n$ variety for $n\ge 4$?

In general this seems like an interesting problem, and one that people may have studied before, but perhaps in some guise that I'm not familiar with, so any input is appreciated.

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