On the number of solutions of \(\sum^{11}_{i=1}\frac {1}{x_i}=1\) in distinct odd natural numbers
The authors study solutions of the Diophantine equation \(\sum_{i=1}^{11} \frac{1}{x_i}=1\), where the \(x_i\) are odd positive integers. It has been mentioned several times in the literature that \(\sum_{i=1}^{k} \frac{1}{x_i}=1\) only has solutions in odd positive integers \(x_i\), if \(k\) is odd, and \(k\geq 9\). To list all 5 solutions with \(k=9\), is an easy exercise, (at least using a computer). When \(k = 11\) there are very many cases to study, and it is not at all easy to enumerate all solutions, as the number of solutions (and the potential search space) is increasing faster than exponentially: \textit{Y. Chen, C. Elsholtz} and \textit{L. Jiang} [Acta Arith. 154, No. 2, 109--123 (2012; Zbl 1271.11041)] proved the number of solutions is \(>c^{k^2}\) for some constant \(c>1\). For \(k=11\), this paper describes a computation to enumerate all 379 118 solutions. The CPU time was about 133 days. The authors also prove some partial results on the case \(k\geq 13\), and also that for \(k \geq 9\) there are solutions of the special type \(\sum_{i=1}^{k-1} \frac{1}{x_i} +\frac{1}{{\text lcm}(x_1, \ldots , x_{k-1})}=1\) in odd positive integers.
scientific article; zbMATH DE number 6154105