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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.
An integral quadratic form has integer coefficients, such as x 2 + xy + y 2; equivalently, given a lattice Λ in a vector space V (over a field with characteristic 0, such as Q or R), a quadratic form Q is integral with respect to Λ if and only if it is integer-valued on Λ, meaning Q(x, y) ∈ Z if x, y ∈ Λ.
The catalecticant of a ternary quartic is the resultant of its 6 second partial derivatives. It vanishes when the ternary quartic can be written as a sum of five 4th powers of linear forms. See also
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]
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 ...
Ramanujan's ternary quadratic form; S. ... Quadratic form (statistics) Surgery structure set; Sylvester's law of inertia; T. Tensor product of quadratic forms; U. U ...
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The oldest problem in the theory of binary quadratic forms is the representation problem: describe the representations of a given number by a given quadratic form f. "Describe" can mean various things: give an algorithm to generate all representations, a closed formula for the number of representations, or even just determine whether any ...