On the number of solutions of the Diophantine equation \(\sum_{i=1}^n\frac{1}{i}=1\) (Q1878826)

For each positive integer \(n\) put \(K(n)=\#\{(x_1,\dots,x_n) : \sum_{i=1}^n {{1}\over {x_i}} = 1\}\), where the \(x_i\) denote positive integers. The author shows that we have \(e^{c{{n^3}\over {\log n}}}<K(n)<c_0^{(1+\varepsilon)2^{n-1}}\) for large \(n\). This result improves the bounds given by Erdős, Graham and Strauss.