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Fermat–Catalan conjecture. In number theory, Fermat's Last Theorem (sometimes called Fermat's conjecture, especially in older texts) states that no three positive integers a, b, and c satisfy the equation an + bn = cn for any integer value of n greater than 2. The cases n = 1 and n = 2 have been known since antiquity to have infinitely many ...
Equation solving. The quadratic formula, the symbolic solution of the quadratic equation ax2 + bx + c = 0. An example of using Newton–Raphson method to solve numerically the equation f(x) = 0. In mathematics, to solve an equation is to find its solutions, which are the values (numbers, functions, sets, etc.) that fulfill the condition stated ...
The same illustration for The midpoint method converges faster than the Euler method, as . Numerical methods for ordinary differential equations are methods used to find numerical approximations to the solutions of ordinary differential equations (ODEs). Their use is also known as "numerical integration", although this term can also refer to ...
The Basel problem is a problem in mathematical analysis with relevance to number theory, concerning an infinite sum of inverse squares. It was first posed by Pietro Mengoli in 1650 and solved by Leonhard Euler in 1734, [1] and read on 5 December 1735 in The Saint Petersburg Academy of Sciences. [2] Since the problem had withstood the attacks of ...
Alternatively, replace the 3n + 1 with n ′ / H(n ′) where n ′ = 3n + 1 and H(n ′) is the highest power of 2 that divides n ′ (with no remainder). The resulting function f maps from odd numbers to odd numbers. Now suppose that for some odd number n, applying this operation k times yields the number 1 (that is, f k (n) = 1).
A recurrence relation is an equation that expresses each element of a sequence as a function of the preceding ones. More precisely, in the case where only the immediately preceding element is involved, a recurrence relation has the form. where. is a function, where X is a set to which the elements of a sequence must belong.
(The fundamental theorem of algebra is the special case n = 1.) This exponential behavior makes solving polynomial systems difficult and explains why there are few solvers that are able to automatically solve systems with Bézout's bound higher than, say, 25 (three equations of degree 3 or five equations of degree 2 are beyond this bound).
An illustration of Newton's method. In numerical analysis, Newton's method, also known as the Newton–Raphson method, named after Isaac Newton and Joseph Raphson, is a root-finding algorithm which produces successively better approximations to the roots (or zeroes) of a real -valued function. The most basic version starts with a real-valued ...