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Displaying similar documents to “In Gaussian integers x 3 + y 3 = z 3 has only trivial solutions – a new approach.”

Solving an indeterminate third degree equation in rational numbers. Sylvester and Lucas

Tatiana Lavrinenko (2002)

Revue d'histoire des mathématiques

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This article concerns the problem of solving diophantine equations in rational numbers. It traces the way in which the 19th century broke from the centuries-old tradition of the purely algebraic treatment of this problem. Special attention is paid to Sylvester’s work “On Certain Ternary Cubic-Form Equations” (1879–1880), in which the algebraico-geometrical approach was applied to the study of an indeterminate equation of third degree.

Division-ample sets and the Diophantine problem for rings of integers

Gunther Cornelissen, Thanases Pheidas, Karim Zahidi (2005)

Journal de Théorie des Nombres de Bordeaux

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We prove that Hilbert’s Tenth Problem for a ring of integers in a number field K has a negative answer if K satisfies two arithmetical conditions (existence of a so-called set of integers and of an elliptic curve of rank one over K ). We relate division-ample sets to arithmetic of abelian varieties.