Bhargava cube

Bhargava cube with the integers a, b, c, d, e, f, g, h at the corners

In mathematics, in number theory, a Bhargava cube (also called Bhargava's cube) is a configuration consisting of eight integers placed at the eight corners of a cube.[1] This configuration was extensively used by Manjul Bhargava, a Canadian-American Fields Medal winning mathematician, to study the composition laws of binary quadratic forms and other such forms. To each pair of opposite faces of a Bhargava cube one can associate an integer binary quadratic form thus getting three binary quadratic forms corresponding to the three pairs of opposite faces of the Bhargava cube.[2] These three quadratic forms all have the same discriminant and Manjul Bhargava proved that their composition in the sense of Gauss[3] is the identity element in the associated group of equivalence classes of primitive binary quadratic forms. (This formulation of Gauss composition was likely first due to Dedekind.)[4] Using this property as the starting point for a theory of composition of binary quadratic forms Manjul Bhargava went on to define fourteen different composition laws using a cube.

  1. ^ Mak Trifkovic (2013). Algebraic Theory of Quadratic Numbers. New York: Springer. p. 175. ISBN 978-1-4614-7716-7.
  2. ^ Manjul Bhargava (2006). Higher composition laws and applications, in Proceedings of the International Congress of Mathematicians, Madrid, Spain, 2006. European Mathematical Society.
  3. ^ Carl Friedrich Gauss (translated by Arthur A Clarke) (1986). Disquisitiones Arithmeticae. Springer Verlag. pp. 230–256.
  4. ^ Richard Dedekind (1932). Gesammelte Mathematische Werke. Vol. 2. Viehweg. p. 307.