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

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