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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
Indian mathematician and astronomer (1114–1185)
William Brouncker in 1657, though his method was more difficult than the chakravala method. The first general method for finding the solutions of the problem
Bhāskara_II
Type of Diophantine equation
equations. Bhaskara II is generally credited with developing the chakravala method, building on the work of Jayadeva and Brahmagupta. Solutions to specific
Pell's_equation
Indian inventions
Apollonius, two of the greatest minds produced by antiquity." Chakravala method – The Chakravala method, a cyclic algorithm to solve indeterminate quadratic equations
List of Indian inventions and discoveries
List_of_Indian_inventions_and_discoveries
to have been made during classical antiquity in Europe. Chakravala method: The Chakravala method, a cyclic algorithm to solve indeterminate quadratic equations
Timeline_of_Indian_innovation
Polynomial equation whose integer solutions are sought
equation in positive integers is x = 226153980, y = 1766319049 (see Chakravala method). In 1900, David Hilbert proposed the solvability of all Diophantine
Diophantine_equation
Development of mathematics in South Asia
of Pell's equation using the chakravala method. The general indeterminate quadratic equation using the chakravala method. Indeterminate cubic equations
Indian_mathematics
Branch of pure mathematics
square root is not rational.) For that matter, the eleventh-century chakravala method amounts—in modern terms—to an algorithm for finding the units of a
Number_theory
solution (including zero and negative solutions) to quadratic equations. Chakravala method, sign convention, madhava series, and the sine and cosine in trigonometric
Indian_people
Pell equations, which he treated using cyclic procedures such as the chakravāla method. In the 14th century, Narayana Pandita completed his Ganita Kaumudi
History_of_mathematics
positive number has two square roots. Furthermore, it also gives the Chakravala method which was the first generalized solution of so-called Pell's equation
Timeline_of_mathematics
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
Indian mathematician and astronomer (598–668)
identity Brahmagupta's formula Brahmagupta theorem Brahmagupta triangle Chakravala method List of Indian mathematicians History of science and technology on
Brahmagupta
Indian mathematician (1892–1953)
publications, including an article on the Chakravala method where he showed how the method differed from the method of continued fractions. He pointed out
A._A._Krishnaswami_Ayyangar
Eratosthenes 263 AD – Gaussian elimination described by Liu Hui 628 – Chakravala method described by Brahmagupta c. 820 – Al-Khawarizmi described algorithms
Timeline_of_algorithms
Efficient way of calculating GCD. Booth's multiplication algorithm Chakravala method: a cyclic algorithm to solve indeterminate quadratic equations, including
List_of_algorithms
apsidal precession of the Sun. 12th century: Bhāskara II develops the Chakravala method, solving Pell's equation. 12th century: Al-Tusi develops a numerical
Timeline of scientific discoveries
Timeline_of_scientific_discoveries
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
List of notable people who belong to the Brahmin caste
Indian mathematician who wrote an article on the difference between Chakravala method and Continued Fractions. Ashutosh Mukherjee, Indian mathematician
List_of_Brahmins
Factorization method based on the difference of two squares
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
Fermat's_factorization_method
Quadratic homogeneous polynomial in two variables
were extended to a complete solution of Pell's equation known as the chakravala method, attributed to either of the Indian mathematicians Jayadeva or Bhāskara
Binary_quadratic_form
Algorithm in number theory
In 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
Treatise on algebra by Bhāskara II
quadratic equations, including Pell's equation which is known as chakravala method or cyclic method. Bijaganita is the first text to recognize that a positive
Bijaganita
11th-century Indian mathematician
was an Indian mathematician, who further developed the cyclic method (Chakravala method) that was called by Hermann Hankel "the finest thing achieved
Jayadeva_(mathematician)
Special-purpose algorithm for factoring integers
An improved method to estimate the probability of success given B1, B2 is found in Kruppa (2010). In practice, the elliptic curve method is faster than
Pollard's_p_−_1_algorithm
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
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
1356 mathematical treatise by Narayana Pandita
examples. Quadratic. 17 rules and 10 examples. Includes a variant of the Chakravala method. Ganita Kaumudi contains many results from continued fractions. In
Ganita_Kaumudi
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
Algorithm for integer factorization
Lenstra elliptic-curve factorization or the elliptic-curve factorization method (ECM) is a fast, sub-exponential running time, algorithm for integer factorization
Lenstra elliptic-curve factorization
Lenstra_elliptic-curve_factorization
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
Algorithm for solving the discrete logarithm problem
It is a fairly simple modification of trial multiplication, the naive method of finding discrete logarithms. Given a cyclic group G {\displaystyle G}
Baby-step_giant-step
Algorithm in computational number theory
<- bk − ⌊μk,j⌉bj; Update B* and the related μi,j's as needed. (The naive method is to recompute B* whenever bi changes: B* <- GramSchmidt({b1, ..., bn})
Lenstra–Lenstra–Lovász lattice basis reduction algorithm
Lenstra–Lenstra–Lovász_lattice_basis_reduction_algorithm
Integer factorization algorithm
to find a nontrivial factor even when n is composite. In that case, the method can be tried again, using a starting value of x other than 2 ( 0 ≤ x < n
Pollard's_rho_algorithm
Products of numbers of the form a^2 + n b^2 are also of that form
obtained. The general method for solving the Pell equation given by Bhaskara II in 1150, namely the chakravala (cyclic) method, was also based on this
Brahmagupta's_identity
Integer factorization algorithm
Probably the most elegant method is to check whether ⌊n1/b⌋b = n holds for any 1 < b ≤ log2(n) using an integer version of Newton's method for the root extraction
Rational_sieve
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
Integer factorization algorithm
exponentially with the digits of the number. Even so, this is a quite satisfactory method, considering that even the best-known algorithms have exponential time growth
Trial_division
Multiplication algorithm
asymptotically fastest multiplication method known from 1971 until 2007. It is asymptotically faster than older methods such as Karatsuba and Toom–Cook multiplication
Schönhage–Strassen_algorithm
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
Method in number theory
probabilistic method of finding roots of polynomials over the field F p {\displaystyle \mathbb {F} _{p}} with p {\displaystyle p} elements. The method was discovered
Berlekamp–Rabin_algorithm
Factorization algorithm
many practical situations, leading to the development of better methods. One such method was suggested by Murphy and Brent; they introduce a two-part score
General_number_field_sieve
Expression of a product of sums of squares as a sum of squares
obtained. The general method for solving the Pell equation given by Bhaskara II in 1150, namely the chakravala (cyclic) method, was also based on this
Brahmagupta–Fibonacci identity
Brahmagupta–Fibonacci_identity
Decomposition of a number into a product
small integer n using mental or pen-and-paper arithmetic, the simplest method is trial division: checking if the number is divisible by prime numbers
Integer_factorization
Special-purpose integer factorization algorithm
root Cipolla Pocklington's Tonelli–Shanks Berlekamp Other algorithms Chakravala Cornacchia Exponentiation by squaring Integer square root Integer relation
Special_number_field_sieve
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
Algorithm for computing greatest common divisors
mathematics, the Euclidean algorithm, or Euclid's algorithm, is an efficient method for computing the greatest common divisor (GCD) of two integers, the largest
Euclidean_algorithm
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
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
Algorithm for generating prime numbers
double-and-increment step. Whenever Eratosthenes' method would cross out k different multiples of a prime 2i + 1, Sundaram's method crosses out i + j(2i + 1) for 1 ≤
Sieve_of_Sundaram
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
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
Integer factorization algorithm
an integer factorization algorithm and, in practice, the second-fastest method known (after the general number field sieve). It is still the fastest for
Quadratic_sieve
Methods to test or prove primality
primality proving (ECPP), are among the quickest and most widely used methods in primality proving. It is an idea put forward by Shafi Goldwasser and
Elliptic_curve_primality
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
Ancient algorithm for generating prime numbers
prime numbers less than or equal to a given integer n by Eratosthenes's method: Create a list of consecutive integers from 2 through n: (2, 3, 4, ...,
Sieve_of_Eratosthenes
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
Mathematical algorithm
prime factor of n {\displaystyle n} . Pollard, J. M. (1978). "Monte Carlo methods for index computation (mod p)". Mathematics of Computation. 32 (143): 918–924
Pollard's rho algorithm for logarithms
Pollard's_rho_algorithm_for_logarithms
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
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
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
Probabilistic primality test
running much faster. It is also slower in practice than commonly used proof methods such as APR-CL and ECPP which give results that do not rely on unproven
Miller–Rabin_primality_test
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
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 computing logarithms
unknown digit in the exponent, and computing that digit by elementary methods. (Note that for readability, the algorithm is stated for cyclic groups
Pohlig–Hellman_algorithm
Problem of inverting exponentiation in groups
logarithms are quickly computable in a few special cases. However, no efficient method is known for computing them in general. In cryptography, the computational
Discrete_logarithm
Algorithm for determining whether a number is prime
hdl:1887/2136. JSTOR 2007581. Riesel, Hans (1994). Prime Numbers and Computer Methods for Factorization. Birkhäuser. pp. 131–136. ISBN 978-0-8176-3743-9. APR
Adleman–Pomerance–Rumely primality test
Adleman–Pomerance–Rumely_primality_test
Algorithms to generate prime numbers
the mainstream method is to generate random numbers in a target range and test them for primality using fast probabilistic methods: a short round of
Generation_of_primes
Study of algorithms for performing number theoretic computations
also known as algorithmic number theory, is the study of computational methods for investigating and solving problems in number theory and arithmetic
Computational_number_theory
Integer factorization algorithm
p+1 method of factoring", Mathematics of Computation, 39 (159): 225–234, doi:10.2307/2007633, JSTOR 2007633, MR 0658227 P + 1 factorization method at Prime
Williams's_p_+_1_algorithm
Probabilistic algorithm for computing discrete logarithms
can be solved faster than with generic methods. The algorithms are indeed adaptations of the index calculus method. Likewise, there’s no known algorithms
Index_calculus_algorithm
Hypothetical world power structure
to establish it as a world religion. Also, the first references to a Chakravala Chakravartin (an emperor who rules over all four of the continents) appears
World_domination
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
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
Primality test for certain numbers
take u0 = 5778. An alternative method for finding the starting value u0 is given in Rödseth 1994. The selection method is much easier than that used by
Lucas–Lehmer–Riesel_test
Probabilistic primality test
select an a (greater than 1 and smaller than n): 47. Using an efficient method for raising a number to a power (mod n) such as binary exponentiation, we
Solovay–Strassen primality test
Solovay–Strassen_primality_test
Algorithm for computing the greatest common divisor
of two numbers was known in ancient China, under the Han dynasty, as a method to reduce fractions: If possible halve it; otherwise, take the denominator
Binary_GCD_algorithm
Quantum algorithm for integer factorization
{\displaystyle \log _{2}(N)} roots of N {\displaystyle N} , e.g., with the Newton method and checking each integer result for primality (AKS primality test). Ekerå
Shor's_algorithm
Algorithm for generating prime numbers
only their values up to N are needed and generated. It remains to find a method for generating the next wheel. Note in the animation that W3 = {1,5,7,11
Sieve_of_Pritchard
Algorithm used in modular arithmetic
Gerhard Rosenberger; Ulrich Hertrampf (24 May 2016). Discrete Algebraic Methods: Arithmetic, Cryptography, Automata and Groups. De Gruyter. pp. 163–165
Tonelli–Shanks_algorithm
Integer factorization algorithm
is a method for integer factorization devised by Daniel Shanks as an improvement on Fermat's factorization method. The success of Fermat's method depends
Shanks's square forms factorization
Shanks's_square_forms_factorization
Algorithm for generating prime numbers
wheel-factorized (2/3/5) sieve of Eratosthenes. To improve its efficiency, a method must be devised to minimize or eliminate these non-productive computations
Sieve_of_Atkin
Primality test for numbers of a certain form
computational work than simple brute force trial division (Schoolhouse method) in the worst case scenario. As 50% of bases a are expected to bear witness
Proth's_theorem
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
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
Number-theoretic algorithm
generator mod N, its order is N − 1 {\displaystyle N-1} and so the method is guaranteed to work for this choice. The above version of Pocklington's
Pocklington_primality_test
CHAKRAVALA METHOD
CHAKRAVALA METHOD
Boy/Male
Tamil
Method, Way, Mode, Manner, One who crosses the river of life, Morning star
Girl/Female
Tamil
Method, Wealth, Protection, Conduct, Auspiciousness, Memory, Well being
Boy/Male
Indian, Sanskrit
Possessor of the Cakra; Worshipper of Vishnu
Male
Greek
(Μεθόδιος) Greek name derived from methodos, METHODIOS means "method."
Boy/Male
Tamil
The scriptures, Vedic method of self realization, Knower of the Vedas, One who knows all, Hindu philosophy or ultimate wisdom, King of all
Girl/Female
Tamil
Method, Wealth, Protection, Conduct, Auspiciousness, Memory, Well being
Boy/Male
Tamil
The scriptures, Vedic method of self realization, Knower of the Vedas, One who knows all, Hindu philosophy or ultimate wisdom, King of all
Boy/Male
Indian, Sanskrit
Circle; Assemblage; A Leader
Boy/Male
Hindu, Indian
A Bird
Boy/Male
Indian, Sanskrit
With a Discus; Emperor
Girl/Female
Tamil
Chairavali | சைராவலீ
Full Moon of Chaitra month
Chairavali | சைராவலீ
Boy/Male
Indian, Sanskrit
Fierce; Forceful; Whirlwind
Boy/Male
Tamil
Vedhanth | வேதாநà¯à®¤
The scriptures, Vedic method of self realization, Knower of the Vedas, One who knows all, Hindu philosophy or ultimate wisdom, King of all
Vedhanth | வேதாநà¯à®¤
Boy/Male
Tamil
The scriptures, Vedic method of self realization, Knower of the Vedas, One who knows all, Hindu philosophy or ultimate wisdom, King of all
Surname or Lastname
English
English : topographic name from Middle English lang, long ‘long’ + strete ‘road’.Translation of Dutch Langestraet, cognate with 1.The confederate general James Longstreet (1821–1904), was born in SC, came from an old Dutch family in New Netherland with the name Langestraet; he was the nephew of Augustus B. Longstreet, a Methodist clergyman born in Augusta, GA, in 1790.
Boy/Male
Tamil
The scriptures, Vedic method of self realization, Knower of the Vedas, One who knows all, Hindu philosophy or ultimate wisdom, King of all
Boy/Male
Tamil
The scriptures, Vedic method of self realization, Knower of the Vedas, One who knows all, Hindu philosophy or ultimate wisdom, King of all
Boy/Male
Tamil
The scriptures, Vedic method of self realization, Knower of the Vedas, One who knows all, Hindu philosophy or ultimate wisdom, King of all
Girl/Female
Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi, Sindhi, Telugu
Full Moon of Chaitra Month
Boy/Male
Indian, Sanskrit
Has a Round Mouth
CHAKRAVALA METHOD
CHAKRAVALA METHOD
Girl/Female
Tamil
Independent, Submissive, Willing, Dependent
Boy/Male
American, British, English
Judge's Son
Female
Slavic
Variant spelling of Slavic Danica, DANIKA means "morning star."
Girl/Female
Indian
Tradition
Surname or Lastname
English
English : habitational name from Hungerford in Berkshire, named with Old English hungor ‘hunger’ (here probably denoting unproductive land) + ford ‘ford’. This surname has been established in Ireland since the 17th century.
Boy/Male
African, Arabic, Australian
Good Man
Boy/Male
Tamil
Fearful
Girl/Female
Tamil
Gyanada | ஜà¯à®žà®¾à®¨à®¾à®¤à®¾
Goddess Saraswati
Boy/Male
Hindu, Indian, Traditional
Courage
Girl/Female
American, Australian, Chinese, French, German, Greek, Latin, Portuguese, Spanish, Swiss, Ukrainian
Healthy; Strong; Strong and Healthy; Brave
CHAKRAVALA METHOD
CHAKRAVALA METHOD
CHAKRAVALA METHOD
CHAKRAVALA METHOD
CHAKRAVALA METHOD
p. pr. & vb. n.
of Methodize
a.
Alt. of Methodical
a.
Of or pertaining to the sect of Methodists; as, Methodist hymns; a Methodist elder.
n.
Classification; a mode or system of classifying natural objects according to certain common characteristics; as, the method of Theophrastus; the method of Ray; the Linnaean method.
n.
One of a sect of Christians, the outgrowth of a small association called the "Holy Club," formed at Oxford University, A.D. 1729, of which the most conspicuous members were John Wesley and his brother Charles; -- originally so called from the methodical strictness of members of the club in all religious duties.
n.
An orderly procedure or process; regular manner of doing anything; hence, manner; way; mode; as, a method of teaching languages; a method of improving the mind.
a.
Of or pertaining to methodists, or to the Methodists.
imp. & p. p.
of Methodize
a.
Alt. of Methodistical
a.
Proceeding with regard to method; systematic.
n.
The science of method or arrangement; a treatise on method.
n.
One who methodizes.
a.
Arranged with regard to method; disposed in a suitable manner, or in a manner to illustrate a subject, or to facilitate practical observation; as, the methodical arrangement of arguments; a methodical treatise.
n.
The act or process of methodizing, or the state of being methodized.
v. t.
To reduce to method; to dispose in due order; to arrange in a convenient manner; as, to methodize one's work or thoughts.
n.
One who observes method.
n.
The art and principles of method.
n.
The system of doctrines, polity, and worship, of the sect called Methodists.
a.
Of or pertaining to the ancient school of physicians called methodists.
a.
Of or pertaining to methodology.