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Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. In mathematics, the Oppenheim conjecture in the area of diophantine approximation concerns representations of numbers by real quadratic forms in several variables. It was formulated in 1929 by Alexander Oppenheim and later the conjectured property was further strengthened by Davenport and Oppenheim. Initial research on this problem took the number n of variables to be large, and applied a version of the Hardy-Littlewood circle method. The definitive work of Margulis, settling the conjecture in the affirmative, used methods arising from ergodic theory and the study of discrete subgroups of semisimple Lie groups.