On sums of seventh powers. (Q1976807)

It is proved that every integer is the sum or difference of 12 integral seventh powers and that every rational member is the sum or difference of 8 seventh powers of rational numbers. A parametric solution in positive integers of the Diophantine equation \(\sum^4_{i=1} x^7_i=\sum^5_{i=1} y^7_i\) is obtained and it is also shown how solutions in positive integers of the equation \(\sum^m_{i=1} x^7_i=\sum^n_{i=1} y^7_i\) may be obtained whenever \(m\geq 4\) and \(n\geq 5\).