1.Introduction

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.


2.Theorem
 
Diophantine 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.
　

3.Example


Case. 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]



4.Reference

[1]: L.J.Mordell, Diophantine Equations, Academic Press, 1969, p. 58.