Find integers $(w, x, y, z)$ such that the product of each two of them minus 1 is square. In the case of $(5, 442, 541)$, the product of each  two of them minus 1 is a square:
$$5 \times 442 - 1 = 47^2, 5 \times 541 - 1 = 52^2, 442\times541 - 1 = 489^2$$
What are the integer-solutions $(w, x, y, z)$ for the case of four numbers, i.e.
$$w x - 1 = a^2, w y - 1 = b^2, w  z - 1 = c^2$$
$$x y - 1 = d^2, x  z - 1 = e^2, y z - 1 = f^2$$
where $a, b, c, d, e, f$ are integers?
 A: The question is a special case of the problem which is labelled diophantine m-tuples in the literature, or more specifically you are asking for a diophantine quadruple with the property D(-1). Andrej Dujella has published numerous papers on the subject, and has a web page here on the subject, which puts the question into that context.
The question is still unsolved, but Dujella et. al. proves in their paper  Effective solution of the D(-1)-quadruple conjecture that there can be only finitely many with $10^{10^{23}}$ being an upper bound on the maximum of $w, x, y , z$ in your notation.
A: Nicolae Ciprian Bonciocat, Mihai Cipu, and Maurice Mignotte, There is no Diophantine $D(-1)$-quadruple, Journal of the London Mathematical Society, Volume 105, Issue 1, January 2022, Pages 63-99, https://doi.org/10.1112/jlms.12507 prove that there are no four distinct positive integers such that each pair has product one more than a square. Here is the abstract:
A set of positive integers with the property that the product of any two of them is the successor of a perfect square is called Diophantine $D(-1)$-set. Such objects are usually studied via a system of generalized Pell equations naturally attached to the set under scrutiny. In this paper, an innovative technique is introduced in the study of Diophantine $D(-1)$-quadruples. The main novelty is the uncovering of a quadratic equation relating various parameters describing a hypothetical $D(-1)$-quadruple with integer entries. In combination with extensive computations, this idea leads to the confirmation of the conjecture according to which there is no Diophantine $D(-1)$-quadruples.
Currently, access at https://londmathsoc.onlinelibrary.wiley.com/doi/full/10.1112/jlms.12507 requires payment or institutional subscription. A version is freely available on arXiv at https://arxiv.org/abs/2010.09200
