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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.
Qalculate! supports common mathematical functions and operations, multiple bases, autocompletion, complex numbers, infinite numbers, arrays and matrices, variables, mathematical and physical constants, user-defined functions, symbolic derivation and integration, solving of equations involving unknowns, uncertainty propagation using interval arithmetic, plotting using Gnuplot, unit and currency ...
Integer factorization algorithms include the Elliptic Curve Method, the Quadratic sieve and the Number field sieve. Algebraic number theory; Magma includes the KANT computer algebra system for comprehensive computations in algebraic number fields. A special type also allows one to compute in the algebraic closure of a field. Module theory and ...
SageMath is designed partially as a free alternative to the general-purpose mathematics products Maple and MATLAB. It can be downloaded or used through a web site. SageMath comprises a variety of other free packages, with a common interface and language. SageMath is developed in Python.
The notion of a quadratic space is a coordinate-free version of the notion of quadratic form. Sometimes, Q is also called a quadratic form. Two n -dimensional quadratic spaces ( V , Q ) and ( V ′, Q ′) are isometric if there exists an invertible linear transformation T : V → V ′ ( isometry ) such that Q ( v ) = Q ′ ( T v ) for all v ...
This is a category of articles relating to software which can be freely used, copied, studied, modified, and redistributed by everyone that obtains a copy: "free software" or "open source software". Typically, this means software which is distributed with a free software license , and whose source code is available to anyone who receives a copy ...
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.
McCullough and Wade [18] extended this approach, which produces all Pythagorean triples when k > h √ 2 /d: Write a positive integer h as pq 2 with p square-free and q positive. Set d = 2pq if p is odd, or d= pq if p is even. For all pairs (h,k) of positive integers, the triples are given by