As the title states, how would I go about finding the positive integer solutions of
$\frac{1}{x_1}+\frac{1}{x_2}+\cdots+\frac{1}{x_n}+\frac{1}{x_1 x_2 \cdots x_n}=1$?
Thank you for your help.
Edit: I think I've made some progress. I conjecture that you can expand the set {a, a+1, a*(a+1)+1, a*(a+1)(a(a+1)+1)+1, ...} (i.e. multiply all previous solutions and add 1), where a=2, to size n, and that will always be a solution in the case of n variables. This is far from finding all positive integer solutions, though.
Edit 2: For example {2}, {2,3}, {2,3,7}, {2,3,7,43}, {2,3,7,1807}, {2,3,7,43,1807,3263443} are all solutions in the case where $n=$ size of the solution set respectively.