This equation was first studied extensively in ancient 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.

* 628 – Brahmagupta gives methods for calculations of the motions and places of various planets, their rising and setting, conjunctions, and calculations of the solar and lunar eclipses

The Indian mathematician Brahmagupta, in Brahma-Sphuta-Siddhanta ( written in A. D. 628 ), discussed the use of negative numbers to produce the general form quadratic formula that remains in use today.

In 628, the Indian mathematician Brahmagupta wrote Brahmasphutasiddhanta which includes, among many other things, a study of equations of the form.

The main work of Brahmagupta, Brāhmasphuṭasiddhānta (" Correctly Established Doctrine of Brahma "), written c. 628, contains ideas including a good understanding of the mathematical role of zero, rules for manipulating both negative and positive numbers, a method for computing square roots, methods of solving linear and some quadratic equations, and rules for summing series, Brahmagupta's identity, and the Brahmagupta ’ s theorem.

The identity was rediscovered by Brahmagupta ( 598 – 668 ), an Indian mathematician and astronomer, who generalized it and used it in his study of what is now erroneously called Pell's equation.

He and Brahmagupta are one of the most renowned Indian mathematicians who made considerable contributions to the study of fractions.

Brahmagupta solved many Pell equations with this method ; in particular he showed how to obtain solutions starting from an integer solution of for k =± 1, ± 2, or ± 4.

Brahmagupta went on to give a recurrence relation for generating solutions to certain instances of Diophantine equations of the second degree such as ( called Pell's equation ) by using the Euclidean algorithm.

Other mathematics included in Śrīpati's work includes, in particular, rules for the solution of simultaneous indeterminate equations of the first degree that are similar to those given by Brahmagupta

Drawing on Greek, Persian and Indian texts — including those of Pythagoras, Plato, Aristotle, Hippocrates, Euclid, Plotinus, Galen, Sushruta, Charaka, Aryabhata and Brahmagupta — the scholars accumulated a great collection of world knowledge, and built on it through their own discoveries.

Drawing on Persian, Indian and Greek texts — including those of Pythagoras, Plato, Aristotle, Hippocrates, Euclid, Plotinus, Galen, Sushruta, Charaka, Aryabhata and Brahmagupta — the scholars accumulated a great collection of knowledge in the world, and built on it through their own discoveries.

The first general method for solving the Pell equation ( for all N ) was given by Bhaskara II in 1150, extending the methods of Brahmagupta.

In 1657, Fermat attempted the Diophantine equation 61x < sup > 2 </ sup > + 1 = y < sup > 2 </ sup > ( solved by Brahmagupta over 1000 years earlier ).

Brahmagupta created a general way to solve Pell's equation known as the chakravala method.

Brahmagupta gave the solution of the general linear equation in chapter eighteen of Brahmasphutasiddhanta,

In its original context, Brahmagupta applied his discovery to the solution of Pell's equation, namely x < sup > 2 </ sup > − Ny < sup > 2 </ sup > = 1.

Edward Saxhau stated that " Brahmagupta, it was he who taught Arabs astronomy ", The famous Abbasid caliph Al-Mansur ( 712 – 775 ) founded Baghdad, which is situated on the banks of the Tigris, and made it a center of learning.

In chapter seven of his Brahmasphutasiddhanta, entitled Lunar Crescent, Brahmagupta rebuts the idea that the Moon is farther from the Earth than the Sun, an idea which is maintained in scriptures.

He expounded on the same subjects on which Aryabhata and Brahmagupta contended, but he expressed them more clearly.

Some notable mathematicians include Archimedes of Syracuse, Leonhard Euler, Carl Gauss, Johann Bernoulli, Jacob Bernoulli, Aryabhata, Brahmagupta, Bhaskara II, Nilakantha Somayaji, Omar Khayyám, Muhammad ibn Mūsā al-Khwārizmī, Bernhard Riemann, Gottfried Leibniz, Andrey Kolmogorov, Euclid of Alexandria, Jules Henri Poincaré, Srinivasa Ramanujan, Alexander Grothendieck, David Hilbert, Alan Turing, von Neumann, Kurt Gödel, Joseph-Louis Lagrange, Georg Cantor, William Rowan Hamilton, Carl Jacobi, Évariste Galois, Nikolay Lobachevsky, Rene Descartes, Joseph Fourier, Pierre-Simon Laplace, Alonzo Church, Nikolay Bogolyubov and Pierre de Fermat.

Brahmagupta is believed to have been born in 598 AD in Bhinmal city in the state of Rajasthan of Northwest India.

Brahmagupta was the first to use zero as a number.

Brahmagupta then goes on to give the sum of the squares and cubes of the first n integers.

It is important to note here Brahmagupta found the result in terms of the sum of the first n integers, rather than in terms of n as is the modern practice.

Brahmagupta's Brahmasphuṭasiddhanta is the very first book that mentions zero as a number, hence Brahmagupta is considered as the man who found zero.

Indian mathematician-astronomer Brahmagupta, in his Brahma-Sphuta-Siddhanta, first recognizes gravity as a force of attraction, and briefly describes the law of gravitation.

Here Brahmagupta states that and as for the question of where he did not commit himself .< ref name =" Boyer Brahmagupta p220 "> His rules for arithmetic on negative numbers and zero are quite close to the modern understanding, except that in modern mathematics division by zero is left undefined.

Aditya expressed characteristics of a cyclic quadrilateral, like Brahmagupta did previously.

All Brahmagupta quadrilaterals with sides a, b, c, d, diagonals e, f, area K, and circumradius R can be obtained by clearing denominators from the following expressions involving rational parameters t, u, and v:

Since the Brahmagupta – Fibonacci identity implies that the product of two integers that can be written as the sum of two squares is itself expressible as the sum of two squares, by applying Fermat's theorem to the prime factorization of any positive integer n, we see that if all of n's odd prime factors congruent to 3 modulo 4 occur to an even exponent, it is expressible as a sum of two squares.

Unfortunately, Indian mathematics remained largely unknown in the West until the late eighteenth century ; Brahmagupta and Bhāskara's work was translated into English in 1817 by Henry Colebrooke.

Some of the important contributions made by Brahmagupta in astronomy are: methods for calculating the position of heavenly bodies over time ( ephemerides ), their rising and setting, conjunctions, and the calculation of solar and lunar eclipses.

It was studied by Brahmagupta in the 7th century, as well as by Fermat in the 17th century .</ td >

Aryabhata of Kusumapura developed the place-value notation in the 5th century and a century later Brahmagupta introduced the symbol for zero.

For instance, for, Brahmagupta composed the triple ( since ) with itself to get the new triple.

Aryabhata ( 476 – 550 ), Brahmagupta ( 598 – 668 ), Albumasar and Al-Sijzi, also proposed that the Earth rotates on its axis.

* Brahmagupta writes the Brahmasphutasiddhanta, an early, yet very advanced, math book.

The Brahmasphutasiddhanta of Brahmagupta ( 598 – 668 ) is the earliest known text to treat zero as a number in its own right and to define operations involving zero.

A Brahmagupta quadrilateral is a cyclic quadrilateral with integer sides, integer diagonals, and integer area.

By Brahmagupta, Bhāsakārācārya. Scanned copy of the book at available online at Google Book.

If the and are real numbers, a more elegant proof is available: the identity expresses the fact that the absolute value of the product of two quaternions is equal to the product of their absolute values, in the same way that the Brahmagupta – Fibonacci two-square identity does for complex numbers.

Brahmagupta (; ) ( 598 – 668 AD ) was an Indian mathematician and astronomer who wrote many important works on mathematics and astronomy.

As a result, Brahmagupta is often referred to as Bhillamalacarya, that is, the teacher from Bhillamala.

Although Brahmagupta was familiar with the works of astronomers following the tradition of Aryabhatiya, it is not known if he was familiar with the work of Bhaskara I, a contemporary.

Brahmagupta had a plethora of criticism directed towards the work of rival astronomers, and in his Brahmasphutasiddhanta is found one of the earliest attested schisms among Indian mathematicians.