According to Mordell' book, diophantine equation x^3 + y^3 + z^3 + w^3= n has infinitely many integer solutions. We show equation ax^3 + ay^3 + bz^3 + bw^3 = n has infinitely many integer solutions.1.IntroductionDiophantine equation ax^3 + ay^3 + bz^3 + bw^3 = n has infinitely many integer solutions under the below condition. condition: There is a solution (x,y,z,w)=(p,q,r,s). -9ab(p+q)(r+s)>0, is not a perfect square. p^2<>q^2 , r^2<>s^2. Proof. ax^3 + ay^3 + bz^3 + bw^3 = n...........................................(1) Let x=p+X, y=q-X, z=r+Y, w=s-Y. (3ap+3aq)X^2+(3ap^2-3aq^2)X+(3br+3bs)Y^2+(3br^2-3bs^2)Y=0...............(2) Equation (1) must satisfy the following conditions. D=-(3ap+3aq)(3br+3bs)=-9ab(p+q)(r+s) >0 Discriminant=(3ap+3aq)(3br^2-3bs^2)^2+(3br+3bs)(3ap^2-3aq^2)^2 <>0......(3) From equation (3), we obtain p^2<>q^2, r^2<>s^2. Q.E.D. @2.Theorem3.ExampleCase. 2x^3+2y^3+z^3+w^3 = 3[x,y,z,w]=[7, -8, 5, 6] [1246, -1247, 537, -526] [1708, -1709, -723, 734] [491113, -491114, 209417, -209406] [673141, -673142, -287023, 287034] [193497472, -193497473, 82507605, -82507594] [265216042, -265216043, -113088495, 113088506][1]: L.J.Mordell, Diophantine Equations, Academic Press, 1969, p. 58.4.Reference

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