-27x^6 - y^6 + z^6 = w^2 Previously, we studied the parametric solutions of ax^6 + by^6 + cz^6 = w^2 below. ax^6 + by^6 + cz^6 = w^2 Ⅰ ax^6 + by^6 + cz^6 = w^2 Ⅱ We show diophantine equation -27x^6 - y^6 + z^6 = w^2 has infinitely many integer solutions. -27x^6 - y^6 + z^6 = w^2....................................................................(1) We use an identity -27(t)^6-(t+1)^6+(4t^2+2t+1)^3 = 9t^2(4t^2+2t+1)(t+1)^2..................(2) Let x=t, y=t+1, z= 4t^2+2t+1, w= 3t(t+1) then we obtain a parametric solution of -27x^6 - y^6 + z^3 = (4t^2+2t+1)w^2. Furthermore, parametric solution of 4t^2+2t+1 =v^2 is given (t,v)=(-2(n-1)/(n^2-4), (n^2-2n+4)/((n-2)(n+2))). n is arbitrary. Thus we obtain a parametric solution -27x^6 - y^6 + z^6 = w^2 as follows. (x,y,z,w)=(2n-2, n^2-2n-2, n^2-2n+4, 6(n-1)(n^2-2n-2)(n^2-2n+4)). Hence equation (1) has infinitely many integer solutions.

HOME