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...

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

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

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

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...

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 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...

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

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

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...

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

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

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

= 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...

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

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

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...

non-performing restoring, non-restoring, and SRT division. Fast division methods start with a close approximation to the final quotient and produce twice...

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

table Pollard, John M. (July 1978) [1977-05-01, 1977-11-18]. "Monte Carlo Methods for Index Computation (mod p)" (PDF). Mathematics of Computation. 32 (143)...

number theory, Springer, 1996. D. Shanks, Class number, a theory of factorization and genera. In Proc. Symp. Pure Math. 20, pages 415—440. AMS, Providence...

optimal strategy for choosing these polynomials is not known; one simple method is to pick a degree d for a polynomial, consider the expansion of n in base...

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

445 The final answer for c is therefore 445, as in the direct method. Like the first method, this requires O(e) multiplications to complete. However, since...

Fermat primes, can be efficiently tested for primality if the prime factorization of p − 1 or p + 1 is known. The sieve of Eratosthenes is generally considered...