On the number of solutions of \({\sum _{j=1}^ s}(1/x_ j)+1/(x_ 1{\cdots}x_ s)=1\) (Q1801577)

The authors study the diophantine equation \[ \sum_{j=1}^ s {1\over {x_ j}}+ {1\over {\prod_{j=1}^ s x_ j}}=1, \qquad 1<x_ 1<\dots< x_ s, \] which has been investigated since 1881 by \textit{J. J. Sylvester} [Am. J. Math. 3, 322-336, 388-390 (1881; JFM 13.0142.02)]. Besides the pure number theory reasons there are motivations from topology, graph theory, group theory, and the area of singularities of algebraic surfaces. \textit{J. Janák} and \textit{L. Skula} [Math. Slovaca 28, 305-310 (1978; Zbl 0418.10001)] published the list of all solutions for \(s\leq 6\) and 18 solutions for \(s=7\). \textit{Z. Cao}, \textit{R. Liu} and \textit{L. Zhang} [J. Number Theory 27, 206-211 (1987; Zbl 0621.10013)] found 5 additional solutions for \(s=7\), and this list was completed by \textit{L. Brenton} and \textit{R. Hill} [Pac. J. Math. 133, 41-67 (1988; Zbl 0616.14032)] to the total of 26 solutions. In the paper of \textit{L. Brenton} and \textit{R. Bruner} [J. Aust. Math. Soc. (to appear)] 42 new solutions of length 8 were otained using the following statement: ``if \(x_ 1,\dots,x_ s\) is a solution of length \(s\), then \(x_ 1,\dots,x_ s\), \(x_ 1\dots x_ s+P\), \(x_ 1\dots x_ s+Q\) is a solution of length \(s+2\) for all factorizations \(PQ\) of \((x_ 1\dots x_ s)^ 2+1\)''. The main result of this paper is given by the presentation of 205 solutions for \(s=9\). The authors use the former statement for factorizations \((x_ 1\dots x_ 7)^ 2+1\), where \(x_ 1,\dots,x_ 7\) are known solutions of length 7. These factorizations are mentioned in Table I.