On sums of eighth powers
This paper deals with the easier Waring problem for eighth powers. Let \(m<n\) be natural numbers, \(a\in\mathbb{Z}\). Consider the equation \[ \sum^ m_{i=1}x_ i^ 8-\sum^ n_{j=1}y^ 8_ j=a. \leqno (1) \] If \(a=0\) and \(m\geq6\), \(n\geq7\) then there exists a solution to (1) in integers depending on a parameter. From this it follows easily that every rational number \(r\) can be written as \(r=\sum^ s_{j=1}\varepsilon_ jx^ 8_ j\) with \(\varepsilon_ j=\pm1\) and \(x_ j\in\mathbb{Q}\), where \(s=12\). Every integer \(r\) can be written in this form with \(x_ j\in\mathbb{Z}\) if \(s=28\). This last result was also obtained by \textit{L. N. Vaserstein} [J. Number Theory 28, 66-68 (1988; Zbl 0634.10042)]. The method also yields infinitely many integer solutions to the system \[ x^ 8_ 1+\cdots+x^ 8_ 7=y^ 8_ 1+\cdots+y^ 8_ 7, x_ 1+\cdots+x_ 7=y_ 1+\cdots+y_ 7. \]
scientific article; zbMATH DE number 14104