<s>[INST] finding all pairs $(x,z)$ How to find  all pairs $(x,z)$ of integers for which $2(z+1)^3$ is divisible by $xz-1$
 [/INST] Here's all the pairs of $(x,z)$ I could find for positive $x,z$.  On the right is $2(z+1)^3/(xz-1)$
\begin{array}{cc}
 \left(1,2\right) & 54 \\
 \left(1,3\right) & 64 \\
 \left(1,5\right) & 108 \\
 \left(1,9\right) & 250 \\
 \left(1,17\right) & 729 \\
 \left(2,1\right) & 16 \\
 \left(2,2\right) & 18 \\
 \left(2,5\right) & 48 \\
 \left(2,14\right) & 250 \\
 \left(3,1\right) & 8 \\
 \left(3,3\right) & 16 \\
 \left(3,11\right) & 108 \\
 \left(3,43\right) & 1331 \\
 \left(5,1\right) & 4 \\
 \left(5,2\right) & 6 \\
 \left(5,5\right) & 18 \\
 \left(5,11\right) & 64 \\
 \left(5,29\right) & 375 \\
 \left(9,1\right) & 2 \\
 \left(9,9\right) & 25 \\
 \left(9,14\right) & 54 \\
 \left(11,3\right) & 4 \\
 \left(11,5\right) & 8 \\
 \left(11,35\right) & 243 \\
 \left(14,2\right) & 2 \\
 \left(14,9\right) & 16 \\
 \left(17,1\right) & 1 \\
 \left(29,5\right) & 3 \\
 \left(29,69\right) & 343 \\
 \left(35,11\right) & 9 \\
 \left(43,3\right) & 1 \\
 \left(69,29\right) & 27
\end{array}
See anything?
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