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  2. Berlekamp's algorithm - Wikipedia

    en.wikipedia.org/wiki/Berlekamp's_algorithm

    By computing the matrix and reducing it to reduced row echelon form and then easily reading off a basis for the null space, we may find a basis for the Berlekamp subalgebra and hence construct polynomials () in it. We then need to successively compute GCDs of the form above until we find a non-trivial factor.

  3. Formula for primes - Wikipedia

    en.wikipedia.org/wiki/Formula_for_primes

    Because the set of primes is a computably enumerable set, by Matiyasevich's theorem, it can be obtained from a system of Diophantine equations. Jones et al. (1976) found an explicit set of 14 Diophantine equations in 26 variables, such that a given number k + 2 is prime if and only if that system has a solution in nonnegative integers: [7]

  4. Factorization of polynomials over finite fields - Wikipedia

    en.wikipedia.org/wiki/Factorization_of...

    In mathematics and computer algebra the factorization of a polynomial consists of decomposing it into a product of irreducible factors.This decomposition is theoretically possible and is unique for polynomials with coefficients in any field, but rather strong restrictions on the field of the coefficients are needed to allow the computation of the factorization by means of an algorithm.

  5. Factorization - Wikipedia

    en.wikipedia.org/wiki/Factorization

    The next odd divisor to be tested is 7. One has 77 = 7 · 11, and thus n = 2 · 3 2 · 7 · 11. This shows that 7 is prime (easy to test directly). Continue with 11, and 7 as a first divisor candidate. As 7 2 > 11, one has finished. Thus 11 is prime, and the prime factorization is; 1386 = 2 · 3 2 · 7 · 11.

  6. Factorization of polynomials - Wikipedia

    en.wikipedia.org/wiki/Factorization_of_polynomials

    A simplified version of the LLL factorization algorithm is as follows: calculate a complex (or p-adic) root α of the polynomial () to high precision, then use the Lenstra–Lenstra–Lovász lattice basis reduction algorithm to find an approximate linear relation between 1, α, α 2, α 3, . . . with integer coefficients, which might be an ...

  7. Fermat's factorization method - Wikipedia

    en.wikipedia.org/wiki/Fermat's_factorization_method

    Fermat's factorization method, named after Pierre de Fermat, is based on the representation of an odd integer as the difference of two squares: N = a 2 − b 2 . {\displaystyle N=a^{2}-b^{2}.} That difference is algebraically factorable as ( a + b ) ( a − b ) {\displaystyle (a+b)(a-b)} ; if neither factor equals one, it is a proper ...

  8. Matrix decomposition - Wikipedia

    en.wikipedia.org/wiki/Matrix_decomposition

    Decomposition: = where C is an m-by-r full column rank matrix and F is an r-by-n full row rank matrix; Comment: The rank factorization can be used to compute the Moore–Penrose pseudoinverse of A, [2] which one can apply to obtain all solutions of the linear system =.

  9. Table of Gaussian integer factorizations - Wikipedia

    en.wikipedia.org/wiki/Table_of_Gaussian_Integer...

    The factorizations are often not unique in the sense that the unit could be absorbed into any other factor with exponent equal to one. The entry 4+2i = −i(1+i) 2 (2+i), for example, could also be written as 4+2i= (1+i) 2 (1−2i). The entries in the table resolve this ambiguity by the following convention: the factors are primes in the right ...