On sums of seventh powers

By means of an identity it is proved that every integer is the sum or diffe rence of 12 integral seventh powers and 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 Sigma(i-1)(4) x(i)(7) = Si gma(i-1)(5) y(t)(7) is obtained and it is also shown how solutions in posit ive integers of the equation Sigma(i-1)(4) x(i)(7) = Sigma(i-1)(5) y(t)(7) obtained whenever m greater than or equal to 4 and n greater than or equal to 5. (C) 2000 Academic Press.