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Algorithm in number theory
number theory, Dixon's factorization method (also Dixon's random squares method or Dixon's algorithm) is a general-purpose integer factorization algorithm;
Dixon's_factorization_method
Factorization method based on the difference of two squares
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 =
Fermat's_factorization_method
Decomposition of a number into a product
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
Integer_factorization
Mathematical for factoring integers
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
Euler's_factorization_method
In number theory, the continued fraction factorization method (CFRAC) is an integer factorization algorithm. It is a general-purpose algorithm, meaning
Continued fraction factorization
Continued_fraction_factorization
Topics referred to by the same term
airports Dixons (Netherlands), a Dutch electricals retailer, originally part of the British Dixons, now independent Dixon's factorization method, an application
Dixons
Algorithm for integer factorization
elliptic-curve factorization or the elliptic-curve factorization method (ECM) is a fast, sub-exponential running time, algorithm for integer factorization, which
Lenstra elliptic-curve factorization
Lenstra_elliptic-curve_factorization
Algorithm for computing greatest common divisors
Lenstra elliptic curve factorization. The Euclidean algorithm may be used to find this GCD efficiently. Continued fraction factorization uses continued fractions
Euclidean_algorithm
Quantum algorithm for integer factorization
circuits. In 2012, the factorization of 15 {\displaystyle 15} was performed with solid-state qubits. Later, in 2012, the factorization of 21 {\displaystyle
Shor's_algorithm
Integer factorization algorithm
factorization is complete. This is roughly the basis of Fermat's factorization method. The quadratic sieve is a modification of Dixon's factorization
Quadratic_sieve
Congruence used in integer factorization algorithms
congruence commonly used in integer factorization algorithms. Given a positive integer n, Fermat's factorization method relies on finding numbers x and y
Congruence_of_squares
Integer factorization algorithm
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
Pollard's_rho_algorithm
Integer having only small prime factors
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
Smooth_number
Special-purpose algorithm for factoring integers
Pollard's p − 1 algorithm is a number theoretic integer factorization algorithm, invented by John Pollard in 1974. It is a special-purpose algorithm,
Pollard's_p_−_1_algorithm
Natural number
{Q} \left[{\sqrt {-n}}\right]} whose ring of integers has a unique factorization, or class number of 1. 9 is the largest single-digit number in the decimal
9
Algorithm for generating numbers coprime with first few primes
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
Wheel_factorization
Factorization algorithm
2007-12-13. "readme.nfs from msieve". "We are pleased to announce the factorization of RSA768, the following 768-bit, 232-digit number from RSA's challenge
General_number_field_sieve
System of rapid mental calculation
calculations that can also be applied to multiplication. The method for general multiplication is a method to achieve multiplications a × b {\displaystyle a\times
Trachtenberg_system
Set of large semiprimes
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
RSA_numbers
Largest integer that divides given integers
= 720. In practice, this method is only feasible for small numbers, as computing prime factorizations takes too long. The method introduced by Euclid for
Greatest_common_divisor
Integer factorization algorithm
square forms factorization is a method for integer factorization devised by Daniel Shanks as an improvement on Fermat's factorization method. The success
Shanks's square forms factorization
Shanks's_square_forms_factorization
Problem of inverting exponentiation in groups
algorithms exist, usually inspired by similar algorithms for integer factorization. These algorithms run faster than the naïve algorithm, some of them
Discrete_logarithm
Method for division with remainder
non-performing restoring, non-restoring, and SRT division. Fast division methods start with a close approximation to the final quotient and produce twice
Division_algorithm
Cyclic algorithm to solve indeterminate quadratic equations
The chakravala method (Sanskrit: चक्रवाल विधि) is a cyclic algorithm to solve indeterminate quadratic equations, including Pell's equation. It is commonly
Chakravala_method
Algorithm for solving the discrete logarithm problem
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
Baby-step_giant-step
Algorithms to generate prime numbers
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
Generation_of_primes
Algorithm in computational number theory
The original applications were to give polynomial-time algorithms for factorizing polynomials with rational coefficients, for finding simultaneous rational
Lenstra–Lenstra–Lovász lattice basis reduction algorithm
Lenstra–Lenstra–Lovász_lattice_basis_reduction_algorithm
Integer factorization algorithm
computational number theory, Williams's p + 1 algorithm is an integer factorization algorithm, one of the family of algebraic-group factorisation algorithms
Williams's_p_+_1_algorithm
Integer factorization algorithm
b2 (mod n), which can be turned into a factorization of n = gcd(a + b, n) × gcd(a − b, n). This factorization might turn out to be trivial (i.e. n = n
Rational_sieve
Multiplication algorithm
π, as well as practical applications such as Lenstra elliptic curve factorization via Kronecker substitution, which reduces polynomial multiplication
Schönhage–Strassen_algorithm
field sieve Lenstra elliptic curve factorization Pollard's p − 1 algorithm Pollard's rho algorithm prime factorization algorithm Quadratic sieve Shor's
List_of_algorithms
Integer factorization algorithm
division is the most laborious but easiest to understand of the integer factorization algorithms. The essential idea behind trial division tests to see if
Trial_division
network-wide coordinate distortion by instead opting for a 3-way factorization. This factorization is as follows: d i , j = Y i ϕ i Y j T {\displaystyle d_{i
Network_Coordinate_System
Integer that is a perfect square modulo some integer
composite moduli whose prime factorization is known. In the case of a composite modulus with unknown prime factorization, the problem of identifying quadratic
Quadratic_residue
Special-purpose integer factorization algorithm
homomorphism φ to the factorization of a+bα, and we can apply the canonical ring homomorphism from Z to Z/nZ to the factorization of a+bm. Setting these
Special_number_field_sieve
Ancient algorithm for generating prime numbers
appears in the original algorithm. This can be generalized with wheel factorization, forming the initial list only from numbers coprime with the first few
Sieve_of_Eratosthenes
Probabilistic primality test
return “composite” return “probably prime” This is not a probabilistic factorization algorithm because it is only able to find factors for numbers n which
Miller–Rabin_primality_test
Exponentation in modular arithmetic
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
Modular_exponentiation
Method for computing the relation of two integers with their greatest common divisor
essential step in the derivation of key-pairs in the RSA public-key encryption method. The standard Euclidean algorithm proceeds by a succession of Euclidean
Extended_Euclidean_algorithm
Algorithm in computational number theory
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)
Pollard's_kangaroo_algorithm
Methods to test or prove primality
Previously-known prime-proving methods such as the Pocklington primality test required at least partial factorization of N ± 1 {\displaystyle N\pm 1}
Elliptic_curve_primality
Probabilistic algorithm for computing discrete logarithms
empty_list for k = 1 , 2 , … {\displaystyle k=1,2,\ldots } Using an integer factorization algorithm optimized for smooth numbers, try to factor g k mod q {\displaystyle
Index_calculus_algorithm
Method in number theory
this polynomial is equivalent to finding its factorization into linear factors. To find such factorization it is sufficient to split the polynomial into
Berlekamp–Rabin_algorithm
Algorithm for integer multiplication
The Toom–Cook algorithm (1963) is a faster generalization of Karatsuba's method, and the Schönhage–Strassen algorithm (1971) is even faster, for sufficiently
Karatsuba_algorithm
Study of algorithms for performing number theoretic computations
for primality testing and integer factorization, finding solutions to diophantine equations, and explicit methods in arithmetic geometry. Computational
Computational_number_theory
Multiplication algorithm
peasant multiplication), one of two multiplication methods used by scribes, is a systematic method for multiplying two numbers that does not require the
Ancient Egyptian multiplication
Ancient_Egyptian_multiplication
Algorithm used in modular arithmetic
composite numbers is a computational problem equivalent to integer factorization. An equivalent, but slightly more redundant version of this algorithm
Tonelli–Shanks_algorithm
Matrix with a multiplicative inverse
above two block matrix inverses can be combined to provide the simple factorization By the Weinstein–Aronszajn identity, one of the two matrices in the
Invertible_matrix
Algorithm for generating prime numbers
odd integer is excluded from the final list if and only if it has a factorization of the form (2i + 1)(2j + 1) — which is to say, if it has a non-trivial
Sieve_of_Sundaram
Algorithm for determining whether a number is prime
JSTOR 2007581. Riesel, Hans (1994). Prime Numbers and Computer Methods for Factorization. Birkhäuser. pp. 131–136. ISBN 978-0-8176-3743-9. APR and APR-CL
Adleman–Pomerance–Rumely primality test
Adleman–Pomerance–Rumely_primality_test
Algorithm for computing logarithms
{\displaystyle g} , an element h ∈ G {\displaystyle h\in G} , and a prime factorization n = ∏ i = 1 r p i e i {\textstyle n=\prod _{i=1}^{r}p_{i}^{e_{i}}}
Pohlig–Hellman_algorithm
Algorithm to multiply two numbers
A multiplication algorithm is an algorithm (or method) to multiply two numbers. Depending on the size of the numbers, different algorithms are more efficient
Multiplication_algorithm
Number-theoretic algorithm
A > N {\displaystyle A>{\sqrt {N}}} , the prime factorization of A is known, but the factorization of B is not necessarily known. If for each prime factor
Pocklington_primality_test
Primality test for certain numbers
March 6, 2016. Riesel, Hans (1994). Prime Numbers and Computer Methods for Factorization. Progress in Mathematics. Vol. 126 (2nd ed.). Birkhäuser. pp. 107–121
Lucas–Lehmer–Riesel_test
Probabilistic primality testing algorithm
Lucas pseudoprimes (with Lucas parameters (P, Q) defined by Selfridge's Method A) are 5459, 5777, 10877, 16109, 18971, 22499, 24569, 25199, 40309, and
Baillie–PSW_primality_test
Mathematical lemma
Bhaskara's Lemma is an identity used as a lemma during the chakravala method. It states that: N x 2 + k = y 2 ⟹ N ( m x + y k ) 2 + m 2 − N k = ( m y +
Bhaskara's_lemma
Audio track separation technique
matrix factorization (NMF) Short-time Fourier transform STFT End-to-end approaches Hybrid approaches Masking-based approaches Repetition-based methods Ensemble
Music_source_separation
Small set of prime numbers used in sieving algorithms
example, Dixon's factorization, the quadratic sieve, and the number field sieve. The difference between these algorithms is essentially the methods used to
Factor_base
Algorithm for generating prime numbers
not outperform a sieve of Eratosthenes with maximum practical wheel factorization (a combination of a 2/3/5/7 sieving wheel and pre-culling composites
Sieve_of_Atkin
Algorithm for multiplying large numbers
Volume 2. Third Edition, Addison-Wesley, 1997. Section 4.3.3.A: Digital methods, pg.294. R. Crandall & C. Pomerance. Prime Numbers – A Computational Perspective
Toom–Cook_multiplication
Mathematical algorithm
problem, analogous to Pollard's rho algorithm to solve the integer factorization problem. The goal is to compute γ {\displaystyle \gamma } such that
Pollard's rho algorithm for logarithms
Pollard's_rho_algorithm_for_logarithms
Probabilistic primality test
more efficiently checked for values of k much smaller than n. (This is the method used by the Great Internet Mersenne Prime Search for testing cofactors.)
Fermat_primality_test
Probabilistic primality test
we know that n is not prime (but this does not tell us a nontrivial factorization of n). This base a is called an Euler witness for n; it is a witness
Solovay–Strassen primality test
Solovay–Strassen_primality_test
Algorithm to solve the discrete logarithm problem
{N} } is a one-way function used in cryptography. Several cryptographic methods are based on the DLP such as the Diffie-Hellman key exchange, the El Gamal
Function_field_sieve
a is a quadratic residue. The algorithm is one of the first efficient methods to solve such a congruence. It was described by H.C. Pocklington in 1917
Pocklington's_algorithm
Field of study
Sayan; Fierer, Noah; David, Lawrence A. (2017-02-09). "Phylogenetic factorization of compositional data yields lineage-level associations in microbiome
Microbial_phylogenetics
Algorithm for computing the greatest common divisor
March 2006). A New GCD Algorithm for Quadratic Number Rings with Unique Factorization. 7th Latin American Symposium on Theoretical Informatics. Valdivia,
Binary_GCD_algorithm
Overview of petroleum in the country
emissions through the elemental carbon fractions and Positive Matrix Factorization method". Environmental Research. 204 (Pt D) 112399. Bibcode:2022ER....20412399D
Oil_in_Turkey
Mathematical procedure
constants. A typical approach in experimental mathematics is to use numerical methods and arbitrary precision arithmetic to find an approximate value for an
Integer_relation_algorithm
Algorithm for generating prime numbers
Pritchard. Sieve of Eratosthenes Sieve of Atkin Sieve theory Wheel factorization Pritchard, Paul (1982). "Explaining the Wheel Sieve". Acta Informatica
Sieve_of_Pritchard
Primality test for numbers of a certain form
ISBN 0-387-94457-5. Hans Riesel (1994). Prime Numbers and Computer Methods for Factorization (2 ed.). Boston, MA: Birkhauser. p. 104. ISBN 3-7643-3743-5. Chris
Proth's_theorem
Fast greatest common divisor algorithm
one digit (in the chosen base, say β = 1000 or β = 232), use some other method, such as the Euclidean algorithm, to obtain the result. If a and b differ
Lehmer's_GCD_algorithm
Formulation of classical mechanics
are the usual starting point for teaching about mechanical systems. This method works well for many problems, but for others the approach is nightmarishly
Lagrangian_mechanics
Efficient algorithm to count points on elliptic curves
) {\displaystyle {\bar {q}}(x,y)} can be done either by double-and-add methods or by using the q ¯ {\displaystyle {\bar {q}}} th division polynomial.
Schoof's_algorithm
Greatest integer less than or equal to square root
{\displaystyle \operatorname {isqrt} (n)} is to use Heron's method, which is a special case of Newton's method, to find a solution for the equation x 2 − n = 0 {\displaystyle
Integer_square_root
Non-homogeneous Poisson process Non-linear least squares Non-negative matrix factorization Nonparametric skew Non-parametric statistics Non-response bias Non-sampling
List_of_statistics_articles
List of terms created from a person's name
Last Theorem, Fermat's little theorem, Fermat's principle, Fermat's factorization method Enrico Fermi, Italian physicist – fermions, Fermi energy, Fermilab
List_of_eponyms_(A–K)
finding such an a {\displaystyle a} , but the following trial and error method can be used. Simply pick an a {\displaystyle a} and by computing the Legendre
Cipolla's_algorithm
Standard division algorithm for multi-digit numbers
(1491) is the earliest printed example of long division, known as the Danda method in medieval Italy, and it became more practical with the introduction of
Long_division
Award in theoretical particle physics
developing concepts and techniques in QCD, such as infrared safety and factorization in hard processes, which permitted precise quantitative predictions
Sakurai_Prize
Kenneth Nordtvedt develops PPN formalism. 1967 – Mendel Sachs publishes factorization of Einstein's field equations. 1967 – Hans Stephani discovers the Stephani
Timeline of gravitational physics and relativity
Timeline_of_gravitational_physics_and_relativity
The University of Mississippi. Retrieved August 17, 2025. "From VPDEI M. Dixon: Celebrating Women's History Month & Dr. Patricia S. Cowings". Shoreline
List of African-American women in STEM fields
List_of_African-American_women_in_STEM_fields
DIXONS FACTORIZATION-METHOD
DIXONS FACTORIZATION-METHOD
Girl/Female
American, Australian, British, Danish, English, German, Greek, Romanian
Divine; From the Sacred Spring; Variant of Dione; Follower of Dionysius
Girl/Female
British, English
A Saxon
Boy/Male
British, English
Dimond
Surname or Lastname
English
English : variant of Ligon with excrescent patronymic -s.
Male
French
French name derived from Latin Dio, a short form of longer names of Greek origin beginning with Dio-, DION means "Zeus."
Boy/Male
British, English
Surname
Female
Hebrew
(דִּימï‹× ָה) Hebrew name DIMONA means "south."
Girl/Female
British, English, Greek, Spanish
From Dionysus God of Wine; Follower of Dionysius
Boy/Male
Teutonic English
Strong leader.
Girl/Female
Celtic
Divine one.
Girl/Female
Spanish
From Dionysus god of wine.
Boy/Male
British, English, French, Greek, Latin
Greek God of Wine
Surname or Lastname
English
English : variant spelling of Dixon.
Surname or Lastname
English and Irish
English and Irish : variant of Diamond.
Girl/Female
American, Australian, British, English, Greek
Divine; Female Version of Dion; Similar to Dennis; Follower of Dionysius
Surname or Lastname
English
English : habitational name from Mixon in Staffordshire, named from Old English mixen ‘dungheap’, or a topographic name for someone who lived by a dungheap.English : patronymic from a pet form of Michael.
Surname or Lastname
English (of Norman origin)
English (of Norman origin) : patronymic from a nickname for a lively person, from Old French hirond, arond ‘swallow’ (the bird).English (of Norman origin) : patronymic from a nickname for a discontented individual, from a diminutive of Old French hire ‘complaint’ (of unknown origin).
Boy/Male
Christian & English(British/American/Australian)
Powerful Ruler
Boy/Male
African, American, Australian, British, English, German, Jamaican, Teutonic
The Legend; Rich; Powerful Ruler
Surname or Lastname
English, North German, and Dutch
English, North German, and Dutch : patronymic from Simon.
DIXONS FACTORIZATION-METHOD
DIXONS FACTORIZATION-METHOD
Girl/Female
Hindu, Indian, Marathi
Fertile and Fragrant Earth
Girl/Female
Greek
Gift.
Surname or Lastname
English
English : habitational name from an unidentified place, perhaps Widefield in Devon or Buckinghamshire, named in Old English with wīd ‘wide’ + feld ‘open country’.
Boy/Male
Gujarati, Indian, Marathi, Punjabi, Sikh
God Rama
Boy/Male
Tamil
Shivangel | ஷீவாநà¯à®•à¯‡à®²Â
Angel messenger of Lord Shiva
Boy/Male
British, English
Son of Gold; Little Golden One
Boy/Male
Biblical
Touching softly, multiplying much.
Female
Hebrew
Variant spelling of Hebrew Channah, CHANAH means "favor; grace."Â
Girl/Female
American, British, English, Latin
Blend of Lily and Elizabeth; The Flower; Innocence; Purity; Beauty; Elizabeth; Fair Lily
Boy/Male
Indian
Little Ali
DIXONS FACTORIZATION-METHOD
DIXONS FACTORIZATION-METHOD
DIXONS FACTORIZATION-METHOD
DIXONS FACTORIZATION-METHOD
DIXONS FACTORIZATION-METHOD
n.
The belief in demons or false gods.
pl.
of Dido
n.
The great hall or council chamber of demons or evil spirits.
a.
Of or pertaining to the Saxons, their country, or their language.
a.
Of or pertaining to a demon or to demons; demoniac.
v. t.
To shackle with irons; to fetter or handcuff.
n.
Specifically: That part of the United States lying north of Mason and Dixon's line. See under Line.
n.
A treatise on demons; a supposititious science which treats of demons and their manifestations.
n.
One who disclaims, disowns, or renounces.
n.
The dominion of demons.
n.
The worship of demons.
n.
The power or government of demons.
a.
Adorned with lions' heads; having arms terminating in lions' heads; -- said of a cross.
n.
A believer in, or worshiper of, demons.
n.
The Greek major third, which comprehend two major tones (the modern major third contains one major and one minor whole tone).
a.
Relating to the Saxons or Anglo- Saxons.
a.
Of or pertaining to the Anglo-Saxons or their language.
n.
One who, or that which, irons.
n.
One in subjection to a demon, or to demons.
n.
The language of the Saxons; Anglo-Saxon.