1.Introduction


We show that X^3 + Y^3 + Z^3 + W^3 = 1 has infinitely many integral solutions.
     
     
2.Theorem
     
Diophantine equation X^3 + Y^3 + Z^3 + W^3 = 1 has infinitely many integral solutions.

X = -(2a+1)(a-1)
Y = -(a+1)(2a-1)
Z = 2a^2
W = 2a^2-1

a is arbitrary.
 
Proof.

Let X = n+a, Y = n+b, Z = -n-c, W = -n...................................(1)
X^3 + Y^3 + Z^3 + W^3....................................................(2)

Substitute (1) to equation (2), we obtain

(3a+3b-3c)n^2+(3a^2+3b^2-3c^2)n+a^3+b^3-c^3..............................(3)

Let n = -(a^2+b^2-c^2)/(a+b-c), c=a+b-1 and b=-a, then we obtain a parametric solution below.

X = -(2a+1)(a-1)
Y = -(a+1)(2a-1)
Z = 2a^2
W = 2a^2-1

Thus X^3 + Y^3 + Z^3 + W^3 = 1 has infinitely many integral solutions.

Q.E.D.