On sums of powers of integers (Q1258328)

Following the method of \textit{K. F. Roth} [J. Lond. Math. Soc. 24, 4-13 (1949; Zbl 0032.01401)], the author proves that for any positive integer \(k\), almost all positive integers \(n\) are expressible in the forms \[ n=x_2^2+x_3^3+x_6^5-x_1^k\quad\text{and}\quad n=x_2^2+x_3^3+x_6^5+x_1^k, \] where the \(x_i\)'s are non-negative integers. Earlier the reviewer [Trans. Am. Math. Soc. 200, 199--205 (1974; Zbl 0292.10043)] showed that sequences of integers expressible in these forms have positive Schnirelman density.