1.Introduction

We show a parametric solution of ax^3 + by^3 + cz^3 + dw^3 = 0 with a+b+c+d <>0.
Thus, diophantine equation ax^3 + by^3 + cz^3 + dw^3 = 0 has infinitely many integer solutions with a+b+c+d <>0.

Parametric solutions of ax^3 + by^3 + cz^3 + dw^3 = 0 are given below.
ax^3 + by^3 + cz^3 + dw^3 = 0 with a+b+c+d = 0.

Parametric solutions of ax^3 + by^3 + cz^3 + dw^3 = 0 Ⅱ


2.Theorem
 
There is a parametric solution of ax^3 + by^3 + cz^3 + dw^3 = 0,

      x = (a^2b+2ab^2)p^3+(-3a^2b-6ab^2)qp^2+(6ab^2+3a^2b)q^2p+(-a^2b-2ab^2)q^3-da^2-2dab-db^2

      y = (-2a^2b-ab^2)p^3+(6a^2b+3ab^2)qp^2+(-3ab^2-6a^2b)q^2p+(2a^2b+ab^2)q^3-da^2-2dab-db^2

      z = (-a^2b+ab^2)p^3+(3a^2b-3ab^2)qp^2+(3ab^2-3a^2b)q^2p+(a^2b-ab^2)q^3+da^2+2dab+db^2

      w = 3(a+b)ab(p-q)^2

a,b,p,q,d: arbitrary
condition: a+b=c.
     
Proof.

ax^3 + by^3 + cz^3 + d =0 ......................................................(1)

Let c = a+b.

Set x=t+p, y=t+q, z=-t+r................................................... ....(2)

(3ap+3bq+3(a+b)r)t^2+(3ap^2+3bq^2-3(a+b)r^2)t+ap^3+bq^3+(a+b)r^3+d = 0..........(3)

Set r = -(ap+bq)/(a+b), then we obtain

t = -1/3(da^2+2dab+db^2+2a^2p^3b+ap^3b^2+bq^3a^2+2b^2q^3a-3a^2p^2bq-3apb^2q^2)/((a+b)ab(p^2+q^2-2pq)).

Substitute r and t to (2), and obtain a parametric solution.

 
Q.E.D.
　

3.Example


Case. x^3 + 2y^3 + 3z^3 + 7w^3 = 0

x = 10p^3-30qp^2+30q^2p-10q^3-63

y = -8p^3+24qp^2-24q^2p+8q^3-63

z = 2p^3-6qp^2+6q^2p-2q^3+63

w = 18(q-p)^2


Case. 3x^3 + 4y^3 + 7z^3 + 11w^3 = 0

x = 132p^3-396qp^2+396q^2p-132q^3-539

y = -120p^3+360qp^2-360q^2p+120q^3-539

z = 12p^3-36qp^2+36q^2p-12q^3+539

w = 252(p-q)^2

p,q: arbitrary