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  2. Collatz conjecture - Wikipedia

    en.wikipedia.org/wiki/Collatz_conjecture

    As an illustration of this, the parity cycle (1 1 0 0 1 1 0 0) and its sub-cycle (1 1 0 0) are associated to the same fraction ⁠ 5 / 7 ⁠ when reduced to lowest terms. In this context, assuming the validity of the Collatz conjecture implies that (1 0) and (0 1) are the only parity cycles generated by positive whole numbers (1 and 2 ...

  3. Equation solving - Wikipedia

    en.wikipedia.org/wiki/Equation_solving

    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 by the equation, consisting generally of two expressions related by an equals sign.

  4. System of polynomial equations - Wikipedia

    en.wikipedia.org/wiki/System_of_polynomial_equations

    Bézout's theorem asserts that a well-behaved system whose equations have degrees d 1, ..., d n has at most d 1 ⋅⋅⋅d n solutions. This bound is sharp. If all the degrees are equal to d, this bound becomes d n and is exponential in the number of variables. (The fundamental theorem of algebra is the special case n = 1.)

  5. Functional equation - Wikipedia

    en.wikipedia.org/wiki/Functional_equation

    Some solutions to functional equations have exploited surjectivity, injectivity, oddness, and evenness. [citation needed] Some functional equations have been solved with the use of ansatzes, mathematical induction. [citation needed] Some classes of functional equations can be solved by computer-assisted techniques. [vague] [4]

  6. System of linear equations - Wikipedia

    en.wikipedia.org/wiki/System_of_linear_equations

    The rank of a system of equations (that is, the rank of the augmented matrix) can never be higher than [the number of variables] + 1, which means that a system with any number of equations can always be reduced to a system that has a number of independent equations that is at most equal to [the number of variables] + 1.

  7. Partial fraction decomposition - Wikipedia

    en.wikipedia.org/wiki/Partial_fraction_decomposition

    In algebra, the partial fraction decomposition or partial fraction expansion of a rational fraction (that is, a fraction such that the numerator and the denominator are both polynomials) is an operation that consists of expressing the fraction as a sum of a polynomial (possibly zero) and one or several fractions with a simpler denominator.

  8. Synthetic division - Wikipedia

    en.wikipedia.org/wiki/Synthetic_division

    In algebra, synthetic division is a method for manually performing Euclidean division of polynomials, with less writing and fewer calculations than long division. It is mostly taught for division by linear monic polynomials (known as Ruffini's rule ), but the method can be generalized to division by any polynomial .

  9. Hilbert's tenth problem - Wikipedia

    en.wikipedia.org/wiki/Hilbert's_tenth_problem

    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.