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

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...

cryptography; Shanks's square forms factorization, integer factorization method that generalizes Fermat's factorization method; and the Tonelli–Shanks algorithm...

it is a proper factorization of N. Each odd number has such a representation. Indeed, if N = c d {\displaystyle N=cd} is a factorization of N, then N =...

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

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...

of squares of the form a2 ≡ b2 (mod N), which can be turned into a factorization of N, N = gcd(a + b, N) × (N/gcd(a + b, N)). This factorization might...

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

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

composite numbers is a computational problem equivalent to integer factorization. An equivalent, but slightly more redundant version of this algorithm...

attempts to set up a congruence of squares modulo n (the integer to be factorized), which often leads to a factorization of n. The algorithm works in two...

as with finding differences of squares in Fermat's factorization method. The great disadvantage of Euler's factorization method is that it cannot be applied...

In number theory, the continued fraction factorization method (CFRAC) is an integer factorization algorithm. It is a general-purpose algorithm, meaning...

as the sum of two squares is itself expressible as the sum of two squares, by applying Fermat's theorem to the prime factorization of any positive integer...

algorithms for integer factorization. These algorithms run faster than the naïve algorithm, some of them proportional to the square root of the size of the...

numbers to form the basis of the factorization wheel. They are known or perhaps determined from previous applications of smaller factorization wheels or...

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

– Factorization of Mersenne numbers Mn (n up to 1280) Factorization of completely factored Mersenne numbers The Cunningham project, factorization of...

integer square root (isqrt) of a non-negative integer n is the non-negative integer m which is the greatest integer less than or equal to the square root...

not assured in arbitrary integral domains. However, if R is a unique factorization domain, then any two elements have a GCD, and more generally this is...

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

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

residues (modulo the number being factorized) in an attempt to find a congruence of squares which will yield a factorization. The number field sieve is the...

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

k primes (cf. wheel factorization), so that the list will start with the next prime, and all the numbers in it below the square of its first element...

Integers of special forms, such as Mersenne primes or Fermat primes, can be efficiently tested for primality if the prime factorization of p − 1 or p + 1...

Formula for primes Factorization RSA number Fundamental theorem of arithmetic Square-free Square-free integer Square-free polynomial Square number Power of...

this speed-up, the number field sieve has to perform computations and factorizations in number fields. This results in many rather complicated aspects of...

using the quarter square method in a digital multiplier. To form the product of two 8-bit integers, for example, the digital device forms the sum and difference...

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