The Mordell-Schinzel conjecture is proven for cubic equations of the form xyz = A(x) + B(y) - c when A and B have degree 3.
Cubic surfaces with infinite, discrete automorphism group
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We prove that the automorphism group of an affine, cubic surface with equation $xyz=g(x,y)$ contains ${\mathbb Z}$ as a finite index subgroup. These equations were first studied by Mordell. v.2: small changes, references updated.
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The Mordell-Schinzel conjecture for cubic diophantine equations
The Mordell-Schinzel conjecture is proven for cubic equations of the form xyz = A(x) + B(y) - c when A and B have degree 3.