On the equation Σ_{j = 1}^s( 1 x_j) + ( 1 (x₁ ... x_s)) = 1 and Znám's problem

We give all the integer solutions of the Diophantine equation Σ_{j = 1}^s( 1 x_j) + ( 1 (x₁ ... x_s)) = 1 (1 < x₁ < ... < x_s) when s = 7 (23 solutions in all). We also prove that (1) if s ≥ 11, then Ω(s + 1) ≥ Ω(s) + 8 and if 2 {does not divide} s, s ≥ 11, then Ω(s + 1) ≥ Ω(s) + 9; (2) If s ≥ 12, then Z(s) ≥ 8, and if 2|s, s ≥ 12, then Z(s) ≥ 9; (3) If s ≥ 10, then A(s + 1) ≥ Ω(s) + Ω(s - 1) + 16, and if 2|s, s ≥ 12, then A(s + 1) ≥ Ω(s) + Ω(s - 1) + 18, where Ω(s), Z(s), and A(s) are defined in the paper.