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Diophantine Geometry

Field: Arithmetic Geometry

The study of the properties of algebraic varieties over fields that are not algebraically closed.

Sequence of Expressions

Expert
No three positive integers a,b,a, b, and cc satisfy the equation\nan+bn=cna^n + b^n = c^n\nfor any integer value of nn greater than 22.
In mathematics, Diophantine geometry is the study of Diophantine equations by means of powerful methods in algebraic geometry. By the 20th century it became clear for some mathematicians that methods of algebraic geometry are ideal tools to study these equations. Diophantine geometry is part of the broader field of arithmetic geometry. Four theorems in Diophantine geometry that are of fundamental importance include: - Mordell–Weil theorem - Roth's theorem - Siegel's theorem - Faltings's theorem - ^Hindry & Silverman 2000, p. vii, Preface. - ^Hindry & Silverman 2000, p. viii, Preface.