On two of John Leech's unsolved problems concerning rational cuboids (Q2786988)

A perfect rational cuboid has integer edge lengths \(X,\) \(Y,\) and \(Z\) such that all face diagonals and the space diagonal are also integers, i.e. there exist integers \(A,\) \(B,\) \(C,\) and \(D\) satisfying NEWLINE\[NEWLINEX^2 + Y^2 = A^2, \quad X^2 + Z^2 = B^2, \quad Y^2 + Z^2 = C^2NEWLINE\]NEWLINE and NEWLINE\[NEWLINEX^2 + Y^2 + Z^2 = D^2.NEWLINE\]NEWLINE If only the first three properties hold, the object is called a \textit{classical rational cuboid}. Infinite families of classical rational cuboids are known, but it is unknown whether perfect rational cuboids exist.NEWLINENEWLINE\textit{J. Leech} [Am. Math. Mon. 84, 518--533 (1977; Zbl 0373.10011)] posed the following related problems. Given a classical rational cuboid: {\parindent=0.7cm\begin{itemize}\item[(1)] Find \(T \in \mathbb{Z}\) such that \(T^2 - X^2 = E^2,\) \(T^2 - Y^2 = F^2,\) and \(T^2 - Z^2 = G^2\) for integers \(E,F,\) and \(G.\) \item[(2)] Find \(T \in \mathbb{Z}\) such that \(T^2 - X^2 - Y^2 = E^2,\) \(T^2 - X^2 - Z^2 = F^2,\) and \(T^2 - Y^2 - Z^2 = G^2\) for integers \(E,F,\) and \(G.\) NEWLINENEWLINE\end{itemize}} If either of these problems does not have a solution, then perfect rational cuboids do not exist.NEWLINENEWLINEIn this paper, the author presents numerical experiments on these problems. Based on suggestive numerical evidence, the author proved that Problem~2 can always be solved, whereas no examples satisfying Problem~1 were found.