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It was originally known as "HECKE and Manin". After a short while it was renamed SAGE, which stands for ‘’Software of Algebra and Geometry Experimentation’’. Sage 0.1 was released in 2005 and almost a year later Sage 1.0 was released. It already consisted of Pari, GAP, Singular and Maxima with an interface that rivals that of Mathematica.
They exist both as "standalone installer" (or "offline installer") & "web installer" (or "online installer"), where the former allows for offline installation as it contains all installation files, whereas the latter needs to download files necessary for installation from the web at the time of installation.
One particular solution is x = 0, y = 0, z = 0. Two other solutions are x = 3, y = 6, z = 1, and x = 8, y = 9, z = 2. There is a unique plane in three-dimensional space which passes through the three points with these coordinates, and this plane is the set of all points whose coordinates are solutions of the equation.
Another ancient problem involving quadratic forms asks us to solve Pell's equation. For instance, we may seek integers x and y so that 1 = x 2 − 2 y 2 {\displaystyle 1=x^{2}-2y^{2}} . Changing signs of x and y in a solution gives another solution, so it is enough to seek just solutions in positive integers.
To find non-primitive solutions (x, y) where gcd(x, y) = g ≠ 1, note that the existence of such a solution implies that g 2 divides m (and equivalently, that if m is square-free, then all solutions are primitive). Thus the above algorithm can be used to search for a primitive solution (u, v) to u 2 + dv 2 = m / g 2 .
For example, the second-order equation y′′ = −y can be rewritten as two first-order equations: y′ = z and z′ = −y. In this section, we describe numerical methods for IVPs, and remark that boundary value problems (BVPs) require a different set of tools. In a BVP, one defines values, or components of the solution y at more than one ...
Hilbert's tenth problem is the tenth on the list of mathematical problems that the German mathematician David Hilbert posed in 1900. It is the challenge to provide a general algorithm that, for any given Diophantine equation (a polynomial equation with integer coefficients and a finite number of unknowns), can decide whether the equation has a solution with all unknowns taking integer values.
Microsoft Mathematics 4.0 (removed): The first freeware version, released in 32-bit and 64-bit editions in January 2011; [8] features a ribbon GUI Microsoft Math for Windows Phone (removed): A branded mobile application for Windows Phone released in 2015 specifically for South African and Tanzanian students; also known as Nokia Mobile ...