number theory, Dixon's factorization method (also Dixon's random squares method or Dixon's algorithm) is a general-purpose integer factorization algorithm;...

and John Brillhart in 1975. The continued fraction method is based on Dixon's factorization method. It uses convergents in the regular continued fraction...

called prime factorization; the result is always unique up to the order of the factors by the prime factorization theorem. To factorize a small integer...

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...

congruence commonly used in integer factorization algorithms. Given a positive integer n, Fermat's factorization method relies on finding numbers x and y...

airports Dixons (Netherlands), a Dutch electricals retailer, originally part of the British Dixons, now independent Dixon's factorization method, an application...

elliptic-curve factorization or the elliptic-curve factorization method (ECM) is a fast, sub-exponential running time, algorithm for integer factorization, which...

Euler's factorization method is a technique for factoring a number by writing it as a sum of two squares in two different ways. For example the number...

step in several integer factorization algorithms, such as Pollard's rho algorithm, Shor's algorithm, Dixon's factorization method and the Lenstra elliptic...

factorization is complete. This is roughly the basis of Fermat's factorization method. The quadratic sieve is a modification of Dixon's factorization...

circuits. In 2012, the factorization of 15 {\displaystyle 15} was performed with solid-state qubits. Later, in 2012, the factorization of 21 {\displaystyle...

a proper subset of the primes as seen in the factor base of Dixon's factorization method and the quadratic sieve. Likewise, it is what the general number...

Pollard's rho algorithm is an algorithm for integer factorization. It was invented by John Pollard in 1975. It uses only a small amount of space, and...

Wheel factorization is a method for generating a sequence of natural numbers by repeated additions, as determined by a number of the first few primes...

Pollard's p − 1 algorithm is a number theoretic integer factorization algorithm, invented by John Pollard in 1974. It is a special-purpose algorithm,...

decimal digits (330 bits). Its factorization was announced on April 1, 1991, by Arjen K. Lenstra. Reportedly, the factorization took a few days using the multiple-polynomial...

integer factorization, primality tests do not generally give prime factors, only stating whether the input number is prime or not. Factorization is thought...

square forms factorization is a method for integer factorization devised by Daniel Shanks as an improvement on Fermat's factorization method. The success...

calculations that can also be applied to multiplication. The method for general multiplication is a method to achieve multiplications a × b {\displaystyle a\times...

algorithms exist, usually inspired by similar algorithms for integer factorization. These algorithms run faster than the naïve algorithm, some of them...

optimal strategy for choosing these polynomials is not known; one simple method is to obtain f from the base-m expansion of n for an appropriate choice...

{Q} \left[{\sqrt {-n}}\right]} whose ring of integers has a unique factorization, or class number of 1. A polygon with nine sides is called a nonagon...

appears in the original algorithm. This can be generalized with wheel factorization, forming the initial list only from numbers coprime with the first few...

= 720. In practice, this method is only feasible for small numbers, as computing prime factorizations takes too long. The method introduced by Euclid for...

The original applications were to give polynomial-time algorithms for factorizing polynomials with rational coefficients, for finding simultaneous rational...

computational number theory, Williams's p + 1 algorithm is an integer factorization algorithm, one of the family of algebraic-group factorisation algorithms...

division is the most laborious but easiest to understand of the integer factorization algorithms. The essential idea behind trial division tests to see if...

return “composite” return “probably prime” This is not a probabilistic factorization algorithm because it is only able to find factors for numbers n which...

Previously-known prime-proving methods such as the Pocklington primality test required at least partial factorization of N ± 1 {\displaystyle N\pm 1}...

The chakravala method (Sanskrit: चक्रवाल विधि) is a cyclic algorithm to solve indeterminate quadratic equations, including Pell's equation. It is commonly...