The equation \(\sum ^9_{i=1} \frac {1}{x_i} = 1\) in distinct odd integers has only the five known solutions (Q2469216)

For the Diophantine equation \(\sum_{i=1}^k \frac1{x_i} = 1\), where\( 1<x_1<x_2< \dots <x_k\), all \(x_i\) are odd, there are, for \(k=9\), according to S. Yamashita [see Rivera and J. Ayala, \url {http://www.primepuzzles.net}], at least five sets of solutions \[ B_1= \{3,5,7,9,11,15,21,231,315\}, \dots, B_5 = \{3,5,7,9,11,15,21,165,693\}. \] The author proves Theorem 1: These five solutions of \(\sum_1^9 \frac1{x_i}=1\) in distinct odd integers are the only possible ones. \textit{``The proof of Theorem 1 is rather long, technical and quite involved.''} First it is shown, that any set \(\{x_1, \dots, x_9\}\) with distinct odd integers, solving (1), contains the set \(T=\{3,5,7,9,11,15\}\). Then it is shown, that the completion of \(T\) to a set of solutions of (1) can be achieved in exactly five ways.