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EDIT. I've just noticed a thread from 2011 in the "Related" column on the right (click me), where a closely related question is being discussed (the main difference seems to be that, in Question 2 below, I'm only asking for the existence of an upper bound).

A subproblem of Problem D18 in the 2004 softcover reprint of the 3rd edition of Guy's Unsolved Problems in Number Theory asks if there are four squares the absolute values of whose differences in pairs are squares (see p. 278 in the book, the subproblem is titled "Four squares whose differences are square"). Guy comments that "Although a solution is unlikely, there do not appear to be any congruence conditions which forbid it." I have two (main) questions:

(Q1) What is the status of the problem? Has there been any substantial progress since 2004?

(Q2) What about the existence of an integer $N \ge 4$ such that there is no $N$-tuple $(a_1, \ldots, a_N)$ of positive integers with the property that $|a_i^2 - a_j^2|$ is a square for all $i, j = 1, \ldots, N$? More precisely, what about $N = 5$?

The question is motivated by a problem in graph theory posed by Sergey Goryainov. I'm mainly looking for references.

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    $\begingroup$ See also this earlier Math Stack Exchange question which has some discussion math.stackexchange.com/questions/4375353/… $\endgroup$
    – JoshuaZ
    Commented Feb 17 at 15:52
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    $\begingroup$ The existence of some upper bound follows from the Bombieri-Lang conjecture - see Noam Elkies' answer on the question you link to, or my answer to mathoverflow.net/questions/408301. $\endgroup$
    – Thomas Bloom
    Commented Feb 18 at 9:53
  • $\begingroup$ @ThomasBloom I see, thanks! Should I conclude that an unconditional proof (of the existence of an upper bound) is hopeless at the current state of the art? Elkies' answer dates back to 2011. Anything new in the meanwhile (and independent from the Bombieri-Lang conjecture)? $\endgroup$
    – Salvo Tringali
    Commented Feb 18 at 10:14

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