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 a Super-$n$ 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: The Super-$n$-variety is 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.
Best Answer
The "Super-$n$" variety, call it $V_n$, seems to be of general type once $n \geq 4$. It probably has no nontrivial rational curves (where "trivial" means that it lies on a hyperplane $c_1=0$ or $c_j=c_k$ some distinct $j,k$; over ${\bf C}$ one must also exclude $c_j=0$ for $j>1$). For $n$ large enough this should follow from the Bombieri-Lang conjectures for the "Super-4" variety $V_4$.
In general if a smooth variety is the "complete intersection" in some projective space ${\bf P}^{N-1}$ of hypersurfaces $P_i=0$ of degrees $d_1,\ldots,d_r$ then it is of general type iff $\sum_i d_i > N$. Here we have $N=(n^2+n)/2$ and $r=(n^2-n)/2$, with each $d_i=2$, so we would get general type once $n\geq 4$; our variety is not quite smooth but the singularities look mild enough not to change the result.
A (very plausible but extremely hard) conjecture of Bombieri and Lang asserts that all the rational points on a variety $X$ of general type lie on a finite union $X_0$ of subvarieties of lower dimension. (This would vastly generalize Faltings' theorems on curves of genus $>1$ [Mordell's conjecture] and subvarieties of abelian varieties.) For complete intersections in ${\bf P}^{N-1}$, the following naïve but suggestive heuristic points in the same way: try the $\sim H^n$ points $(x_1:x_2:...:x_N)$ with integers $x_m$ such that $H \leq \max_m |x_m| < 2H$; for any choice of such $x_m$ we have $|P_i(\vec x)| \ll H^{d_i}$ for each $i$, and if we imagine these $r$ numbers $P_i(\vec x)$ are more-or-less randomly and independently distributed among integers of those sizes then the expected number of $\vec x$ where they're all zero is of order $H^{N-\sum_i d_i}$. This means that the general-type case is precisely when the exponent is negative, and thus that (summing over $H=1,2,4,8,16,\ldots$) the total number of rational points if finite. This heuristic cannot account for non-random rational points due to polynomial identities, but those are precisely the subvarieties that the Bombieri-Lang conjecture allows.
It seems reasonable to guess that already for $n=4$ there are no nontrivial rational curves, and that the nontrivial part of $X_0$ is finite or even empty. Unfortunately, even assuming the B-L conjecture there is no known way to determine $X_0$. Nevertheless it may be possible to deduce that some $V_n$ has no nontrivial points under the assumption that B-L holds for $V_4$. The reason is that for $n > 4$ there are many maps from $V_n$ to $V_4$, obtained by choosing any $4$ of the $n$ variables $c_1,\ldots,c_n$ in order, and the corresponding six $d$'s. If all nontrivial points of $V_4$ were known to lie on a union $X_0$ of proper subvarieties, then any nontrivial point of $V_n$ would have to lie on the intersection of preimages of $X_0$ under $n \choose 4$ different maps, and whatever $X_0$ turns out to be, such an intersection ought to be trivial if $n$ is large enough.
NB it would likely require some nontrivial [sic] work to make a proof of this even assuming the B-L conjecture for $V_4$, but such an analysis was carried out in a similar context in the famous paper
and something like that should be possible here.