[math-fun] System of 3 equations

Do you know if it is possible to solve this system of equations, with n=3 or 4: a^n + b^n = c^n + d^n e^n + f^n = g^n + h^n (ae)^n + (bf)^n = (cg)^n + (dh)^n a,b,c,d,e,f,g,h being distinct integers. Solutions with n=1 or 2 are easy to find. For example: 1 + 4 = 2 + 3 6 + 7 = 5 + 8 (1*6) + (4*7) = (2*5) + (3*8) 2² + 11² = 5² + 10² 79² + 46² = 86² + 31² (2*79)² + (11*46)² = (5*86)² + (10*31)² But what about n=3? I am unable to find at least one solution. And n=4 seems hopeless. Christian.