DIAGONAL CUBIC EQUATIONS IN FOUR VARIABLES WITH PRIME COEFFICIENTS Carmen Laura Basile Department of Mathematics, Imperial College of Science, Technology and Medicine, London (
[email protected])
Thomas Anthony Fisher Department of Pure Mathematics and Mathematical Statistics, Sidney Sussex College, Cambridge (
[email protected])
November 2000
Abstract The aim of this paper is to give an alternative proof of a theorem of R. HeathBrown [3] regarding the existence of non-zero integral solutions of the equation
pX 1
3 1
+
pX 2
3 2
+
pX 3
3 3
+
pX 4
3 4
= 0 , where the
pj are prime integers 2 (mod 3).
We start by presenting the main result of this paper. This result has been proved by Roger Heath-Brown [3] under the conjecture that the di erence s(A) r(A) between the Selmer rank and the arithmetic rank of the elliptic curve X 3 + Y 3 = AZ 3 is even. In this note we will show that we do not need this assumption and give a detailed proof of this result. Theorem 1 Let p1 ; p2 ; p3 ; p4 be prime integers such that pi 2 (mod 3) (1 Then the equation p1 X13 + p2 X23 + p3 X33 + p4 X43 = 0
i 4).
has non-zero integral solutions, assuming the conjecture that the Tate-Shafarevich group of the elliptic curve X 3 + Y 3 = AZ 3 over Q is nite.