Lehmer's GCD algorithm, named after Derrick Henry Lehmer, is a fast GCD algorithm, an improvement on the simpler but slower Euclidean algorithm. It is...

binary GCD algorithm, also known as Stein's algorithm or the binary Euclidean algorithm, is an algorithm that computes the greatest common divisor (GCD) of...

mathematics, the Euclidean algorithm, or Euclid's algorithm, is an efficient method for computing the greatest common divisor (GCD) of two integers, the largest...

compute gcd(48,18), one proceeds as follows: gcd ( 48 , 18 ) → gcd ( 48 − 18 , 18 ) = gcd ( 30 , 18 ) → gcd ( 30 − 18 , 18 ) = gcd ( 12 , 18 ) → gcd ( 12...

States Greatest common divisor Binary GCD algorithm Polynomial greatest common divisor Lehmer's GCD algorithm Griffith College Dublin, in Dublin, Ireland...

algorithm can in turn be run on those until only primes remain. A basic observation is that, using Euclid's algorithm, we can always compute the GCD between...

theory algorithms for multiprecision integers, such as factoring, Euclid's algorithm, long division, and proof of primality, he also formulated Lehmer's conjecture...

extended Euclidean algorithm is an extension to the Euclidean algorithm, and computes, in addition to the greatest common divisor (gcd) of integers a and...

steps: Pseudocode for Pollard's rho algorithm x ← 2 // starting value y ← x d ← 1 while d = 1: x ← g(x) y ← g(g(y)) d ← gcd(|x - y|, n) if d = n: return failure...

Lehmer's Mahler measure problem), a problem in number theory, after Derrick Henry Lehmer Lehmer five, named after Dick Lehmer Lehmer's GCD algorithm,...

g = gcd(aM − 1, n) = 13. Since 1 < 13 < 299, thus return 13. 299 / 13 = 23 is prime, thus it is fully factored: 299 = 13 × 23. Since the algorithm is incremental...

(D/p)=+1} , this algorithm degenerates into a slow version of Pollard's p − 1 algorithm. So, for different values of M we calculate gcd ( N , V M − 2 )...

coprime is given by the Euclidean algorithm and its faster variants such as binary GCD algorithm or Lehmer's GCD algorithm. The number of integers coprime...

Euclidean algorithm. In particular, division by some v mod n {\displaystyle v{\bmod {n}}} includes calculation of the gcd ( v , n ) {\displaystyle \gcd(v,n)}...

{\displaystyle O(n^{2}\log p)} . Taking the gcd {\displaystyle \gcd } of two polynomials via Euclidean algorithm works in O ( n 2 ) {\displaystyle O(n^{2})}...

Schoof's algorithm is an efficient algorithm to count points on elliptic curves over finite fields. The algorithm has applications in elliptic curve cryptography...

Euler–Jacobi pseudoprime. When n is odd and composite, at least half of all a with gcd(a,n) = 1 are Euler witnesses. We can prove this as follows: let {a1, a2,...

In computational number theory, Cornacchia's algorithm is an algorithm for solving the Diophantine equation x 2 + d y 2 = m {\displaystyle x^{2}+dy^{2}=m}...

{p}}+1\right)^{2}\leq \left({\sqrt[{4}]{N}}+1\right)^{2}<q} and thus gcd ( q , m p ) = 1 {\displaystyle \gcd(q,m_{p})=1} and there exists an integer u with the property...

other three numbers in this range, 3, 6, and 9 are not, since gcd(9, 3) = gcd(9, 6) = 3 and gcd(9, 9) = 9. Therefore, φ(9) = 6. As another example, φ(1) =...

k = gcd ⁡ ( a − c , d − b ) {\displaystyle k=\operatorname {gcd} (a-c,d-b)} and h = gcd ⁡ ( a + c , d + b ) {\displaystyle h=\operatorname {gcd} (a+c...

for which all values of a {\displaystyle a} with gcd ⁡ ( a , n ) = 1 {\displaystyle \operatorname {gcd} (a,n)=1} are Fermat liars. For these numbers, repeated...

and n = AB). Hence, if factoring is a goal, these gcd calculations can be inserted into the algorithm at little additional computational cost. This leads...

= gcd ( 194 , 1649 ) ⋅ gcd ( 34 , 1649 ) = 97 ⋅ 17 {\displaystyle 1649=\gcd(194,1649)\cdot \gcd(34,1649)=97\cdot 17} using the Euclidean algorithm to...

"Two Fast GCD Algorithms". Journal of Algorithms. 16 (1): 110–144. doi:10.1006/jagm.1994.1006. Crandall, R.; Pomerance, C. (2005). "Algorithm 9.4.7 (Stehlé-Zimmerman...

(also Dixon's random squares method or Dixon's algorithm) is a general-purpose integer factorization algorithm; it is the prototypical factor base method...

{\displaystyle b} is a primitive root of m {\displaystyle m} and gcd ( a , m ) = 1 {\displaystyle \gcd(a,m)=1} . Discrete logarithms are quickly computable in...

factorization of Δ and by taking a gcd, this ambiguous form provides the complete prime factorization of n. This algorithm has these main steps: Let n be...

Carmichael numbers. However, a slightly weaker variant of the converse is Lehmer's theorem: If there exists an integer a such that a p − 1 ≡ 1 ( mod p ) {\displaystyle...

(1 < gcd(a,n) < n for some a ≤ r), output composite. For (a = r; a > 1; a--) { If ((gcd = GCD[a,n]) > 1 && gcd < n), Return[Composite] } gcd = {GCD(29,31)=1...