Examples of "chakravala"
"Bhaskara's" Lemma is an identity used as a lemma during the chakravala method. It states that:
Ayyangar wrote an article on the Chakravala method and showed how the method differs from the method of continued fractions. He recounted that this point was missed by Andre Weil, who thought that the Chakravala method was only an “experimental fact” to the Indians and attributed general proofs to Fermat and Lagrange.
Jayadeva (9th century) and Bhaskara (12th century) offered the first complete solution to the equation, using the "chakravala" method to find for formula_2 the solution
This case was notorious for its difficulty, and was first solved in Europe by Brouncker in 1657–58 in response to a challenge by Fermat, using continued fractions. A method for the general problem was first completely described rigorously by Lagrange in 1766. Lagrange's method, however, requires the calculation of 21 successive convergents of the continued fraction for the square root of 61, while the "chakravala" method is much simpler. Selenius, in his assessment of the "chakravala" method, states
For that matter, the 11th-century chakravala method amounts—in modern terms—to an algorithm for finding the units of a real quadratic number field. However, neither Bhāskara nor Gauss knew of number fields as such.
"Chakra" in Sanskrit means cycle. As per popular legend, Chakravala indicates a mythical range of mountains which orbits around the earth like a wall and not limited by light and darkness.
Bhaskara derived a cyclic, "chakravala" method for solving indeterminate quadratic equations of the form ax + bx + c = y. Bhaskara's method for finding the solutions of the problem Nx + 1 = y (the so-called "Pell's equation") is of considerable importance.
The "chakravala" method () is a cyclic algorithm to solve indeterminate quadratic equations, including Pell's equation. It is commonly attributed to Bhāskara II, (c. 1114 – 1185 CE) although some attribute it to Jayadeva (c. 950 ~ 1000 CE). Jayadeva pointed out that Brahmagupta's approach to solving equations of this type could be generalized, and he then described this general method, which was later refined by Bhāskara II in his "Bijaganita" treatise. He called it the Chakravala method: "chakra" meaning "wheel" in Sanskrit, a reference to the cyclic nature of the algorithm. C.-O. Selenius held that no European performances at the time of Bhāskara, nor much later, exceeded its marvellous height of mathematical complexity.
Not only did this give a way to generate infinitely many solutions to "x" − "Ny" = 1 starting with one solution, but also, by dividing such a composition by "k""k", integer or "nearly integer" solutions could often be 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 identity.
Not only did this give a way to generate infinitely many solutions to "x" − "Ny" = 1 starting with one solution, but also, by dividing such a composition by "k""k", integer or "nearly integer" solutions could often be 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 identity.
In 1657, Fermat attempted to solve the Diophantine equation (solved by Brahmagupta over 1000 years earlier). The equation was eventually solved by Euler in the early 18th century, who also solved a number of other Diophantine equations. The smallest solution of this equation in positive integers is , (see Chakravala method).
In Bijaganita Bhāskara II refined Jayadeva's way of generalization of Brahmagupta's approach to solving indeterminate 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 number has two square roots
Jayadeva (जयदेव) was a ninth-century Indian mathematician, who further developed the cyclic method (Chakravala method) that was called by Hermann Hankel "the finest thing achieved in the theory of numbers before Lagrange (18th century)". He also made significant contributions to combinatorics.
The first general method for solving the Pell equation (for all "N") was given by Bhaskara II in 1150, extending the methods of Brahmagupta. Called the chakravala (cyclic) method, it starts by composing any triple formula_18 (that is, one which satisfies formula_19) with the trivial triple formula_20 to get the triple formula_21, which can be scaled down to
Brahmagupta (628 CE) started the systematic study of indefinite quadratic equations—in particular, the misnamed Pell equation, in which Archimedes may have first been interested, and which did not start to be solved in the West until the time of Fermat and Euler. Later Sanskrit authors would follow, using Brahmagupta's technical terminology. A general procedure (the chakravala, or "cyclic method") for solving Pell's equation was finally found by Jayadeva (cited in the eleventh century; his work is otherwise lost); the earliest surviving exposition appears in Bhāskara II's Bīja-gaṇita (twelfth century).
Indian people have played a major role in the development of the philosophy, sciences, mathematics, arts, architecture and astronomy throughout history. During ancient period, notable mathematics accomplishment of India included Hindu–Arabic numeral system with decimal place-value and a symbol for zero, interpolation formula, fibonacci's identity, theorem, the first "complete" arithmetic solution (including zero and negative solutions) to quadratic equations. Chakravala method, sign convention, madhava series, and the sine and cosine in trigonometric functions can be traced to the "jyā" and "koti-jyā". Notable military inventions include war elephants, crucible steel weapons popularly known as Damascus steel and Mysorean rockets. Other notable inventions during ancient period include chess, cotton, sugar, fired bricks, carbon pigment ink, ruler, lac, lacquer, stepwell, indigo dye, snake and ladder, muslin, ludo, calico, Wootz steel, incense clock, shampoo, palampore, chintz, and prefabricated home.
This equation was first studied extensively in India, starting with Brahmagupta, who developed the "chakravala" method to solve Pell's equation and other quadratic indeterminate equations in his "Brahma Sphuta Siddhanta" in 628, about a thousand years before Pell's time. His "Brahma Sphuta Siddhanta" was translated into Arabic in 773 and was subsequently translated into Latin in 1126. Bhaskara II in the 12th century and Narayana Pandit in the 14th century both found general solutions to Pell's equation and other quadratic indeterminate equations. Solutions to specific examples of the Pell equation, such as the Pell numbers arising from the equation with "n" = 2, had been known for much longer, since the time of Pythagoras in Greece and to a similar date in India. The name of Pell's equation arose from Leonhard Euler's mistakenly attributing Lord Brouncker's solution of the equation to John Pell.
There is circumstantial evidence of protohistoric knowledge of algebraic identities involving binary quadratic forms. The first problem concerning binary quadratic forms asks for the existence or construction of representations of integers by particular binary quadratic forms. The prime examples are the solution of Pell's equation and the representation of integers as sums of two squares. Pell's equation was already considered by the Indian mathematician Brahmagupta in the 7th century CE. Several centuries later, his ideas 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 II. The problem of representing integers by sums of two squares was considered in the 6th century by Diophantus. In the 17th century, inspired while reading Diophantus's Arithmetica, Fermat made several observations about representations by specific quadratic forms including that which is now known as Fermat's theorem on sums of two squares. Euler provided the first proofs of Fermat's observations and added some new conjectures about representations by specific forms, without proof.