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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.
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.
In linear algebra and projective geometry, Gerbaldi's theorem, proved by Gerbaldi , states that one can find six pairwise apolar linearly independent nondegenerate ternary quadratic forms. These are permuted by the Valentiner group.
For example, in section V, article 303, Gauss summarized his calculations of class numbers of proper primitive binary quadratic forms, and conjectured that he had found all of them with class numbers 1, 2, and 3. This was later interpreted as the determination of imaginary quadratic number fields with even discriminant and class number 1, 2 ...
Pages in category "Quadratic forms" ... Generalized Clifford algebra; Genus of a quadratic form; ... Ramanujan's ternary quadratic form; S. Signature (topology) ...
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 ...
Sylvester’s inequality about the rank (linear algebra) of the product of two matrices. Sylvester's closed solution for the Frobenius coin problem when there are only two coins. Sylvester's triangle problem, a particular geometric representation of the sum of three vectors of equal length
The theorem states that if the quadratic form defines a homomorphism into the positive real numbers on the non-zero part of the algebra, then the algebra must be isomorphic to the real numbers, the complex numbers, the quaternions, or the octonions, and that there are no other possibilities.
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