On the equation \(\sum ^{s}_{j=1}(1/x_ j)+(1/(x_ 1\,\dots \,x_ s))=1\) and Znám's problem (Q1090359)

Consider the equations \[ (1)\quad \sum^{s}_{j=1}(1/x_ j)+(1/(x_ 1...x_ s))=1,\quad 1<x_ 1<x_ 2<...<x_ s \] and \[ (2)\quad \sum^{s}_{j=1}(1/x_ j)-(1/(x_ 1...x_ s))=1,\quad 1<x_ 1<x_ 2<...<x_ s,\quad s>2. \] Znám asked whether there exists an integer \(x_ i\) for every integer \(s>1\) such that \(x_ i\) is a proper factor of \(x_ 1...x_{i-1}x_{i+1}...x_ s+1\) for \(i=1,...,s.\) Let \(\Omega\) (s) be the number of solutions of (1), Z(s) be the number of solutions of Znám's problem and A(s) be the number of solutions of (2). In this paper, the authors give all the 23 integer solutions of (1) when \(s=7\). They also prove that (i) if \(s\geq 11\), then \(\Omega (s+1)\geq \Omega (s)+8\) and if 2 \(\nmid s\), \(s\geq 11\), then \(\Omega (s+1)\geq \Omega (s)+9;\) (ii) if \(s\geq 12\), then Z(s)\(\geq 8\) and if 2 \(| s\), \(s\geq 12\), then Z(s)\(\geq 9;\) (iii) if \(s\geq 10\), then \(A(s+1)\geq \Omega (s)+\Omega (s-1)+16\) and if 2 \(| s\), \(s\geq 12\), then \(A(s+1)\geq \Omega (s)+\Omega (s-1)+18\).