On the power values of power sums

Using several currently available techniques, including Baker's method, Frey curves and modular forms, we prove that for odd values of k   with 1⩽k<1701⩽k<170, the equationk1+k2+⋯+xk=y2n1k+2k+⋯+xk=y2n in positive integers x,y,nx,y,n with n>2n>2 has only the trivial solution (x,y)=(1,1)(x,y)=(1,1).