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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
Simple continued fraction whose partial denominators repeat periodically
In mathematics, an infinite periodic continued fraction is a simple continued fraction that can be placed in the form x = a 0 + 1 a 1 + 1 a 2 + 1 ⋱ a
Periodic_continued_fraction
Decomposition of a number into a product
factorization be solved in polynomial time on a classical computer? More unsolved problems in computer science In mathematics, integer factorization is
Integer_factorization
Algorithm for computing greatest common divisors
curve factorization. The Euclidean algorithm may be used to find this GCD efficiently. Continued fraction factorization uses continued fractions, which
Euclidean_algorithm
(Mathematical) decomposition into a product
example, 3 × 5 is an integer factorization of 15, and (x − 2)(x + 2) is a polynomial factorization of x2 − 4. Factorization is not usually considered meaningful
Factorization
Quantum algorithm for integer factorization
qubits. 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
Procedure to solve equations of second degree
analytical theory of continued fractions. Here is a simple example to illustrate the solution of a quadratic equation using continued fractions. We begin with
Solving quadratic equations with continued fractions
Solving_quadratic_equations_with_continued_fractions
Number divisible only by 1 and itself
hold for unique factorization domains. The fundamental theorem of arithmetic continues to hold (by definition) in unique factorization domains. An example
Prime_number
American mathematician (1930–2022)
in integer factorization. His joint work with Michael A. Morrison in 1975 describes how to implement the continued fraction factorization method originally
John_Brillhart
American mathematician (1905–1991)
studied physics and earned a bachelor's degree from UC Berkeley, and continued with graduate studies at the University of Chicago. He and his father
D._H._Lehmer
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
Bonse's inequality Prime factor Table of prime factors Formula for primes Factorization RSA number Fundamental theorem of arithmetic Square-free Square-free
List_of_number_theory_topics
American mathematician (1875–1952)
American Mathematical Society, Vol. 40, No. 12 (1934), p. 883 Continued fraction factorization R. E. Powers (1911). "The Tenth Perfect Number". American Mathematical
Ralph_Ernest_Powers
Positive real number which when multiplied by itself gives 5
integral domain that is not a unique factorization domain. For example, the number 6 has two inequivalent factorizations within this ring: 6 = 2 ⋅ 3 = ( 1
Square_root_of_5
Base-12 numeral system
2 appears twice in the factorization of twelve, whereas only once in the factorization of ten; which means that most fractions whose denominators are
Duodecimal
Integer factorization algorithm
Shanks' 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
Type of Diophantine equation
convergents of a continued fraction share the same property: If pk−1/qk−1 and pk/qk are two successive convergents of a continued fraction, then the matrix
Pell's_equation
Natural number
equations Diophantine approximation Irrationality measure Simple continued fractions Category List of topics List of recreational topics Wikibook Wikiversity
1
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
Congruence_of_squares
Cryptographic attack on the RSA system
Wiener, is a type of cryptographic attack against RSA. The attack uses continued fraction representation to expose the private key d when d is small. Fictional
Wiener's_attack
Base sixty numeral system
sexagesimal system, any fraction in which the denominator is a regular number (having only 2, 3, and 5 in its prime factorization) may be expressed exactly
Sexagesimal
Unique positive real number which when multiplied by itself gives 2
based on the sequence of Pell numbers, which can be derived from the continued fraction expansion of 2 {\displaystyle {\sqrt {2}}} . Despite having a smaller
Square_root_of_2
Mathematical concept
{\displaystyle S_{c}} in a continued fraction corresponds to reducing the quadratic form. The eventually periodic nature of the continued fraction is then reflected
Quadratic_irrational_number
Number whose square is a given number
primes having an odd power in the factorization are necessary. More precisely, the square root of a prime factorization is p 1 2 e 1 + 1 ⋯ p k 2 e k + 1
Square_root
Function that is holomorphic on the whole complex plane
particular, if for meromorphic functions one can generalize the factorization into simple fractions (the Mittag-Leffler theorem on the decomposition of a meromorphic
Entire_function
the polynomial and its derivative. The square-free factorization of a polynomial p is a factorization p = p 1 p 2 2 ⋯ p k k {\displaystyle p=p_{1}p_{2}^{2}\cdots
Polynomial_root-finding
American mathematician (1867–1938)
professor at Berkeley in 1918 and continued to teach there until retiring in 1937. In 1903, he presented a factorization of Jevons's number (8,616,460,799)
Derrick_Norman_Lehmer
Branch of pure mathematics
composite numbers. Factorization is a method of expressing a number as a product. Specifically in number theory, integer factorization is the decomposition
Number_theory
Number-theoretical function
_{d\mid n \atop d\,\equiv \,1,3{\pmod {4}}}(-1)^{(d-1)/2}} The prime factorization n = 2 g p 1 f 1 p 2 f 2 ⋯ q 1 h 1 q 2 h 2 ⋯ {\displaystyle
Sum_of_squares_function
Result in modular arithmetic
factors modulo p into two coprime polynomials, this factorization can be lifted to a factorization modulo any higher power of p (the case of roots corresponds
Hensel's_lemma
Methods for locating real roots of a polynomial
reasons for that. Firstly Yun's algorithm for computing the square-free factorization is less costly than twice the cost of the computation of the greatest
Real-root_isolation
Extension of the factorial function
"Exponential integral E: Continued fraction representations (Formula 06.34.10.0005)". "Exponential integral E: Continued fraction representations (Formula
Gamma_function
Integer that is a perfect square modulo some integer
quadratic reciprocity. Several modern factorization algorithms (including Dixon's algorithm, the continued fraction method, the quadratic sieve, and the
Quadratic_residue
Way to break a division problem into smaller steps
{){\color {White}0}95}}\\\ \ \ 19\\\end{array}}} This results in the factorization 950 = 2 × 5 2 × 19 {\displaystyle 950=2\times 5^{2}\times 19} . When
Short_division
Turing-complete esoteric programming language invented by John Conway
positive fractions together with an initial positive integer input n. The program is run by updating the integer n as follows: for the first fraction f in
FRACTRAN
Number that is not a ratio of integers
contradictory pair of prime factorizations and hence violates the fundamental theorem of arithmetic (unique prime factorization). A stronger result is the
Irrational_number
^{2}EI}{(KL)^{2}}}} Euler's continued fraction formula connecting a finite sum of products with a finite continued fraction Euler product formula for the
List of topics named after Leonhard Euler
List_of_topics_named_after_Leonhard_Euler
some series of positive integers (counting numbers). It is a sum of unit fractions. If infinitely many numbers have their reciprocals summed, generally the
List_of_sums_of_reciprocals
Sum of inverse squares of natural numbers
the sine function as an infinite product is valid, by the Weierstrass factorization theorem), but even without justification, by simply obtaining the correct
Basel_problem
1356 mathematical treatise by Narayana Pandita
nx^{2}+k^{2}=y^{2}} . Contains factorization method, 11 rules and 7 examples. Contains rules for writing a fraction as a sum of unit fractions. 22 rules and 14 examples
Ganita_Kaumudi
American mathematician
While not entirely rigorous, his proof that Morrison and Brillhart's continued fraction factoring algorithm ran in roughly e 2 ln n ln ln n {\displaystyle
Richard_Schroeppel
Natural number
factor of 99999. the smallest integer whose square root has a simple continued fraction with period 3. a prime index prime, as 13 is prime. In Mexico "cuarenta
41_(number)
Method for representing or encoding numbers
introduced fractions to Islamic countries in the early 9th century; his fraction presentation was similar to the traditional Chinese mathematical fractions from
Positional_notation
Japanese manga artist
Sexual Desires (正しい変態性欲) Delusions That Jump Out at You: Kago Shintaro's Factorization Eccentric People Magazine (奇人画報) Health Plan (健康の設計) Industrial Revolution
Shintaro_Kago
Natural number
292 is a repdigit in base 8 with it being 444. In the simplified continued fraction for pi, 292 is the 5th number. "Facts about the integer". mathworld
292_(number)
Cross-platform reverse-Polish calculator program
3, 2019. "Advanced Bash-Scripting Guide, Chapter 16, Example 16-52 (Factorization)". Retrieved 2020-09-20. Adam Back. "Diffie–Hellman in 2 lines of Perl"
Dc_(computer_program)
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
Symbols used in mathematical expressions
tals_of_Mathematics_(Burzynski_and_Ellis)/03%3A_Exponents_Roots_and_Factorization_of_Whole_Numbers/3.02%3A_Grouping_Symbols_and_the_Order_of_Operations
Symbols_of_grouping
Polynomial equation of degree two
given a quadratic equation in the form x2 + bx + c = 0, the sought factorization has the form (x + q)(x + s), and one has to find two numbers q and s
Quadratic_equation
Number, approximately 2.41421
from this an infinite number of further relations can be found. Continued fraction pattern of a few low powers σ − 1 = [ 0 ; 2 , 2 , 2 , 2 , . . . ]
Silver_ratio
How many times a number is divisible by 2
an integer. This is equivalent to the multiplicity of 2 in the prime factorization. A singly even number can be divided by 2 only once; it is even but
Singly_and_doubly_even
Algorithm for finding polynomial roots
type is straightforward, whereas addition is performed following the factorization c 3 = c 1 + c 2 = | c 1 | ⋅ ( α 1 + α 2 | c 2 | | c 1 | ) {\displaystyle
Graeffe's_method
Base-20 numeral system
as 10 (two and five), a fraction will terminate in decimal if and only if it terminates in vigesimal. The prime factorization of twenty is 22 × 5, so
Vigesimal
Natural number
2016-05-31. "Sloane's A086383 : Primes found among the denominators of the continued fraction rational approximations to sqrt(2)". The On-Line Encyclopedia of Integer
29_(number)
Rational numbers with root 5 added
and a unique factorization domain. Any positive element of the golden field can be written as a generalized type of continued fraction, in which the
Golden_field
Computer algebra system
Michael Lucks; Bruce W. Char (1986). A fast implementation of polynomial factorization | Proceedings of SYMSAC '86. ACM. pp. 228–232. ISBN 978-0-89791-199-3
Axiom (computer algebra system)
Axiom_(computer_algebra_system)
Tool for solving polynomial equations
field of formal Laurent series in the indeterminate X, i.e. the field of fractions of the formal power series ring K [ [ X ] ] {\displaystyle K[[X]]} , over
Newton_polygon
Proof that a number is prime
that problems such as primality testing and the complement of integer factorization lie in NP, the class of problems verifiable in polynomial time given
Primality_certificate
Eratosthenes Sieve of Pritchard Sieve of Sundaram Wheel factorization Integer factorization Continued fraction (CFRAC) Dixon's Lenstra elliptic curve (ECM) Euler's
Quadratic_Frobenius_test
Numbers obtained by adding the two previous ones
unimodular matrix. This property can be understood in terms of the continued fraction representation for the golden ratio φ: φ = 1 + 1 1 + 1 1 + 1 1 + ⋱
Fibonacci_sequence
Lattice network
are not possible, so some form of approximation has to be used. A continued fraction expansion of tanh(x) is tanh ( x ) = 1 1 x + 1 3 x + 1 5 x + 1 7
Lattice_delay_network
Infinitely many prime numbers exist
fundamental theorem of arithmetic: that every integer has a unique prime factorization. What Euler wrote (not with this modern notation and, unlike modern
Euclid's_theorem
Mathematical constant
{arccot}(55)+\operatorname {arccot}(14187)-\cdots }).} The simple continued fraction expansion is OEIS: A016730 ln 2 = [ 0 ; 1 , 2 , 3 , 1 , 6 , 3 ,
Natural_logarithm_of_2
Coincidence in mathematics
irrational values. These are explainable in terms of large terms in the continued fraction representation of the irrational value, but further insight into why
Mathematical_coincidence
Transformation of a mathematical sequence
its connection to the continued fraction representation of a number. Let 0 < x < 1 {\displaystyle 0<x<1} have the continued fraction representation x = [
Binomial_transform
Counting real roots of a polynomial in an interval
Hidulph Vincent. Roughly speaking, Vincent's theorem consists of using continued fractions for replacing Budan's linear transformations of the variable by Möbius
Budan's_theorem
Branch of elementary mathematics
focuses on their properties and relationships such as divisibility, factorization, and primality. Traditionally, it is known as higher arithmetic. Numbers
Arithmetic
1-factorable. The perfect 1-factorization conjecture that every complete graph on an even number of vertices admits a perfect 1-factorization. Cereceda's conjecture
List of unsolved problems in mathematics
List_of_unsolved_problems_in_mathematics
Natural number
The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. "Continued Fractions and Characteristic Recurrences". Math Pages. Sloane, N. J. A. (ed
6000_(number)
Overview of petroleum in the country
non-exhaust traffic emissions through the elemental carbon fractions and Positive Matrix Factorization method". Environmental Research. 204 (Pt D) 112399. Bibcode:2022ER
Oil_in_Turkey
Natural number
square where the digits are squares. In this case, 4 and 9 are squares. The fraction 1/49 is a repeating decimal with a period of 42: 1/49 = 0.0204081632
49_(number)
Polynomial root-finding algorithm
non-constant c i ( m ) {\textstyle c_{i}(m)} . Hence the limit of the fraction is the same as that of c 1 ( m + 1 ) c 1 ( m ) , {\textstyle {\frac
Bernoulli's_method
1955 mathematics book by Constance Reid
connects these topics back to the integers through the theory of continued fractions and the prime number theorem. The final chapter, Chapter ℵ 0 {\displaystyle
From_Zero_to_Infinity
Display device
such as computed tomography and non-negative matrix factorization and non-negative tensor factorization. Each of these display technologies can be seen to
3D_display
Density-based data clustering algorithm
Sibylle; Morik, Katharina (2018). The Relationship of DBSCAN to Matrix Factorization and Spectral Clustering (PDF). Lernen, Wissen, Daten, Analysen (LWDA)
DBSCAN
single valid codeword from the received word could not tolerate a greater fraction of errors. This resulted in a gap between the error-correction performance
List_decoding
German polymath and scholar (1777–1855)
deals with infinite continued fractions arising as ratios of hypergeometric functions, which are now called Gauss continued fractions. In 1823, Gauss won
Carl_Friedrich_Gauss
Composite number which passes Miller–Rabin primality test
Miller–Rabin primality test. All prime numbers pass this test, but a small fraction of composites also pass, making them "pseudoprimes". Unlike the Fermat
Strong_pseudoprime
Algorithm for computing the greatest common divisor
variant of Lehmer's GCD algorithm, and the relationship between GCD and continued fraction expansions of real numbers. Vallée, Brigitte (September–October 1998)
Binary_GCD_algorithm
"The Erdös-Moser equation 1k + 2k +...+ (m−1)k = mk revisited using continued fractions". Mathematics of Computation. 80. American Mathematical Society:
List_of_prime_numbers
Equation for radii of tangent circles
common divisor. Every primitive root quadruple can be found from a factorization of a sum of two squares, n 2 + m 2 = d e {\displaystyle n^{2}+m^{2}=de}
Descartes'_theorem
Two raised to an integer power
in the OEIS) By comparison, powers of two with negative exponents are fractions: for positive integer n, 2−n is one half multiplied by itself n times
Power_of_two
Analytic function in mathematics
1016/S0377-0427(02)00358-8. MR 1906742. Cvijović, Djurdje; Klinowski, Jacek (1997). "Continued-fraction expansions for the Riemann zeta function and polylogarithms". Proc
Riemann_zeta_function
Power series with negative powers
example, consider the following rational function, along with its partial fraction expansion: f ( z ) = 1 ( z − 1 ) ( z − 2 i ) = 1 + 2 i 5 ( 1 z − 1 − 1
Laurent_series
Arithmetic operation, inverse of nth power
Newton's method can be modified to produce various generalized continued fractions for the nth root. For example,[citation needed] z n = x n + y n =
Nth_root
Complex-differentiable (mathematical) function
mean holomorphic except at certain isolated poles), resembles a rational fraction ("part") of entire functions in a domain of the complex plane. Cauchy had
Holomorphic_function
Study of numbers that are not solutions of polynomials with rational coefficients
papers on the matter in the 1840s sketched out arguments using simple continued fractions to construct transcendental numbers. Later, in the 1850s, he gave
Transcendental_number_theory
Type of mathematical integrals
representation of it is shown below. Fully expanded, this value turns into a fraction that involves two 2736 digit integers. π 2 ( 1 − 3 ⋅ 5 ⋯ 113 ⋅ ( 1 / 3
Borwein_integral
Natural number
number (19) of fourth powers. 239/169 is a convergent of the simple continued fraction of the square root of 2, so that 2392 = 2 · 1692 − 1. Related to the
239_(number)
Geometric representation of the complex numbers
necessary, and not just convenient. Consider the infinite periodic continued fraction f ( z ) = 1 + z 1 + z 1 + z 1 + z ⋱ . {\displaystyle f(z)=1+{\cfrac
Complex_plane
Attribute of a mathematical function
Mittag-Leffler's theorem Methods of contour integration Morera's theorem Partial fractions in complex analysis Ahlfors, Lars (1979). Complex Analysis. McGraw Hill
Residue_(complex_analysis)
Estimate of time taken for running an algorithm
sub-exponential time algorithm is the best-known classical algorithm for integer factorization, the general number field sieve, which runs in time about 2 O ( n 1
Time_complexity
Mathematical concept in polynomial theory
S_{k}(r)^{k}=\operatorname {res} _{r}(rQ'(x)-P(x),Q(x))} be the square-free factorization of the resultant which appears on the right. Trager proved that the
Resultant
approximation. Additionally, since the limiting part of the computation is the factorization of the admittance matrix and this is done only once, its performance
Holomorphic Embedding Load-flow method
Holomorphic_Embedding_Load-flow_method
Automated information retrieval method
component analysis Semi-discrete decomposition Non-negative matrix factorization Singular value decomposition Matrix decomposition techniques are data-driven
Concept_search
Silverman–Toeplitz theorem (mathematical analysis) Śleszyński–Pringsheim theorem (continued fraction) Stolz–Cesàro theorem (calculus) Stone–Weierstrass theorem (functional
List_of_theorems
learning about real numbers and basic number theory (prime numbers, prime factorization, fundamental theorem of arithmetic, ratios, and percentages), topics
Mathematics education in the United States
Mathematics_education_in_the_United_States
German mathematician (1882–1935)
Fields, 1927) characterized the rings in which the ideals have unique factorization into prime ideals (now called Dedekind domains). Noether showed that
Emmy_Noether
Theorem in number theory
the k primes p1, ..., pk can show up (with exponent 1) in the prime factorization of r, there are at most 2k different possibilities for r. Furthermore
Divergence of the sum of the reciprocals of the primes
Divergence_of_the_sum_of_the_reciprocals_of_the_primes
Group-like structure appearing in global fields
between reducing binary quadratic forms and continued fraction expansion; one step in the continued fraction expansion of a certain quadratic irrationality
Infrastructure (number theory)
Infrastructure_(number_theory)
CONTINUED FRACTION-FACTORIZATION
CONTINUED FRACTION-FACTORIZATION
Girl/Female
Muslim
Contended
Girl/Female
Arabic, Muslim
Continues
Boy/Male
Hindu
Tone continued, Not final
Girl/Female
Celtic
Contented.
Girl/Female
Bengali, Indian
Fraction of Time
Boy/Male
Indian
Friction
Girl/Female
Hindu, Indian
Contented
Girl/Female
Australian, Celtic
Contented
Girl/Female
Muslim
Contended
Boy/Male
Hindu, Indian
Tone Continued
Boy/Male
African, Ghana, Hindu, Indian
Forceful; Long; Continued Beauty
Boy/Male
Arabic, Indian, Muslim
Contented
Girl/Female
Hindu, Indian
Fraction of the Cosmos
Girl/Female
Arabic, Muslim, Sindhi
Contented
Boy/Male
Hindu, Indian
Continuer
Girl/Female
Indian
Contended
Boy/Male
Indian
Contended
Boy/Male
Tamil
Anantim | அநாநà¯à®¤à®¿à®®
Tone continued, Not final
Anantim | அநாநà¯à®¤à®¿à®®
Girl/Female
Arabic, Muslim, Pashtun
Contended
Boy/Male
British, English, Indian, Russian
Work
CONTINUED FRACTION-FACTORIZATION
CONTINUED FRACTION-FACTORIZATION
Boy/Male
Arabic, Muslim, Pashtun
Jewellery for the Nose
Girl/Female
Hindu, Indian
God
Boy/Male
British, English
From the Cottage by the Wall
Girl/Female
Tamil
Cool
Boy/Male
Arthurian Legend
A knight.
Girl/Female
Australian, Biblical, French, Greek, Jamaican
Supplying
Boy/Male
Indian, Punjabi, Sikh
Pure Love
Girl/Female
Bengali, Hindu, Indian, Japanese, Kannada, Malayalam, Marathi, Sindhi, Tamil, Telugu
Flower
Boy/Male
Hindi
A storm god.
Boy/Male
Arabic
Servant of the merciful.
CONTINUED FRACTION-FACTORIZATION
CONTINUED FRACTION-FACTORIZATION
CONTINUED FRACTION-FACTORIZATION
CONTINUED FRACTION-FACTORIZATION
CONTINUED FRACTION-FACTORIZATION
n.
The adhesive friction of a wheel on a rail, a rope on a pulley, or the like.
v. t.
To retain; to suffer or cause to remain; as, the trustees were continued; also, to suffer to live.
a.
Apt to break out into a passion; apt to scold; cross; snappish; ugly; unruly; as, a fractious man; a fractious horse.
v. t.
To separate by means of, or to subject to, fractional distillation or crystallization; to fractionate; -- frequently used with out; as, to fraction out a certain grade of oil from pretroleum.
v. i.
To deliver an oration.
n.
The appropriate action of any special organ or part of an animal or vegetable organism; as, the function of the heart or the limbs; the function of leaves, sap, roots, etc.; life is the sum of the functions of the various organs and parts of the body.
n.
One who, or that which, continues; esp., one who continues a series or a work; a continuer.
n.
The mutual or reciprocal action of chemical agents upon each other, or the action upon such chemical agents of some form of energy, as heat, light, or electricity, resulting in a chemical change in one or more of these agents, with the production of new compounds or the manifestation of distinctive characters. See Blowpipe reaction, Flame reaction, under Blowpipe, and Flame.
a.
Uninterrupted; unbroken; continual; continued.
imp. & p. p.
of Continue
n.
Friction.
n.
Basso continuo, or continued bass.
n.
The act of drawing, or the state of being drawn; as, the traction of a muscle.
p. p. & a.
Having extension of time, space, order of events, exertion of energy, etc.; extended; protracted; uninterrupted; also, resumed after interruption; extending through a succession of issues, session, etc.; as, a continued story.
n.
Previous action.
n.
One who continues; one who has the power of perseverance or persistence.
a.
Of or pertaining to fractions or a fraction; constituting a fraction; as, fractional numbers.
v. i.
To be steadfast or constant in any course; to persevere; to abide; to endure; to persist; to keep up or maintain a particular condition, course, or series of actions; as, the army continued to advance.
a.
Prolonged; continued.
n.
Any action in resisting other action or force; counter tendency; movement in a contrary direction; reverse action.