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In number theory, a branch of mathematics, Ramanujan's ternary quadratic form is the algebraic expression x 2 + y 2 + 10z 2 with integral values for x, y and z. [1] [2] Srinivasa Ramanujan considered this expression in a footnote in a paper [3] published in 1916 and briefly discussed the representability of integers in this form.
Pierre de Fermat gave a criterion for numbers of the form 8a + 1 and 8a + 3 to be sums of a square plus twice another square, but did not provide a proof. [1] N. Beguelin noticed in 1774 [2] that every positive integer which is neither of the form 8n + 7, nor of the form 4n, is the sum of three squares, but did not provide a satisfactory proof. [3]
Noether, Emmy (1908), "Über die Bildung des Formensystems der ternären biquadratischen Form (On Complete Systems of Invariants for Ternary Biquadratic Forms)", Journal für die reine und angewandte Mathematik, 134: 23–90 and two tables, archived from the original on 2013-03-08.
A mapping q : M → R : v ↦ b(v, v) is the associated quadratic form of b, and B : M × M → R : (u, v) ↦ q(u + v) − q(u) − q(v) is the polar form of q. A quadratic form q : M → R may be characterized in the following equivalent ways: There exists an R-bilinear form b : M × M → R such that q(v) is the associated quadratic form.
Quadratic form; R. Ramanujan's sum; Ramanujan's ternary quadratic form; S. Square (algebra) Square number; Sum of squares function; Sum of two squares theorem; Sums ...
Gauss contributed to solving the Kepler conjecture in 1831 with the proof that a greatest packing density of spheres in the three-dimensional space is given when the centres of the spheres form a cubic face-centred arrangement, [127] when he reviewed a book of Ludwig August Seeber on the theory of reduction of positive ternary quadratic forms ...
A quadratic form with integer coefficients is called an integral binary quadratic form, often abbreviated to binary quadratic form. This article is entirely devoted to integral binary quadratic forms. This choice is motivated by their status as the driving force behind the development of algebraic number theory.
Pages in category "Quadratic forms" ... Generalized Clifford algebra; Genus of a quadratic form; ... Ramanujan's ternary quadratic form; S. Signature (topology) ...
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