Quadratic theory?

Question

Who came up with the quadratic theory and why?

Answer 1

History

Vedic number theory
Mathematicians in India were interested in finding integral solutions of Diophantine equations since the Vedic era. The earliest geometric use of Diophantine equations can be traced back to the Sulba Sutras, which were written between the 8th and 6th centuries BC. Baudhayana (c. 800 BC) found two sets of positive integral solutions to a set of simultaneous Diophantine equations, and also used simultaneous Diophantine equations with up to four unknowns. Apastamba (c. 600 BC) used simultaneous Diophantine equations with up to five unknowns.


Jaina number theory
In India, Jaina mathematicians developed the earliest systematic theory of numbers from the 4th century BC to the 2nd century CE. The Jaina text Surya Prajinapti (c. 400 BC) classifies all numbers into three sets: enumerable, innumerable and infinite. Each of these was further subdivided into three orders:

Enumerable: lowest, intermediate and highest.
Innumerable: nearly innumerable, truly innumerable and innumerably innumerable.
Infinite: nearly infinite, truly infinite, infinitely infinite.
The Jains were the first to discard the idea that all infinites were the same or equal. They recognized five different types of infinity: infinite in one and two directions (one dimension), infinite in area (two dimensions), infinite everywhere (three dimensions), and infinite perpetually (infinite number of dimensions).

The highest enumerable number N of the Jains corresponds to the modern concept of aleph-null (the cardinal number of the infinite set of integers 1, 2, ...), the smallest cardinal transfinite number. The Jains also defined a whole system of transfinite cardinal numbers, of which is the smallest.

In the Jaina work on the theory of sets, two basic types of transfinite numbers are distinguished. On both physical and ontological grounds, a distinction was made between asmkhyata and ananata, between rigidly bounded and loosely bounded infinities.


Hellenistic number theory
Number theory was a favorite study among the Hellenistic mathematicians of Alexandria, Egypt from the 3rd century CE, who were aware of the Diophantine equation concept in numerous special cases. The first Hellenistic mathematician to study these equations was Diophantus.

Diophantus also looked for a method of finding integer solutions to linear indeterminate equations, equations that lack sufficient information to produce a single discrete set of answers. The equation x + y = 5 is such an equation. Diophantus discovered that many indeterminate equations can be reduced to a form where a certain category of answers is known even though a specific answer is not.


Classical Indian number theory
Diophantine equations were extensively studied by mathematicians in medieval India, who were the first to systematically investigate methods for the determination of integral solutions of Diophantine equations. Aryabhata (499) gave the first explicit description of the general integral solution of the linear Diophantine equation ay + bx = c, which occurs in his text Aryabhatiya. This kuttaka algorithm is considered to be one of the most significant contributions of Aryabhata in pure mathematics, which found solutions to Diophantine equations by means of continued fractions. The technique was applied by Aryabhata to give integral solutions of simulataneous linear Diophantine equations, a problem with important applications in astronomy. He also found the general solution to the indeterminate linear equation using this method.

Brahmagupta in 628 handled more difficult Diophantine equations. He used the chakravala method to solve quadratic Diophantine equations, including forms of Pell's equation, such as 61x2 + 1 = y2. His Brahma Sphuta Siddhanta was translated into Arabic in 773 and was subsequently translated into Latin in 1126. The equation 61x2 + 1 = y2 was later posed as a problem in 1657 by the French mathematician Pierre de Fermat. The general solution to this particular form of Pell's equation was found over 70 years later by Leonhard Euler, while the general solution to Pell's equation was found over 100 years later by Joseph Louis Lagrange in 1767. Meanwhile, many centuries ago, the general solution to Pell's equation was recorded by Bhaskara II in 1150, using a modified version of Brahmagupta's chakravala method, which he also used to find the general solution to other indeterminate quadratic equations and quadratic Diophantine equations. Bhaskara's chakravala method for finding the general solution to Pell's equation was much simpler than the method used by Lagrange over 600 years later. Bhaskara also found solutions to other indeterminate quadratic, cubic, quartic and higher-order polynomial equations. Narayana Pandit further improved on the chakravala method and found more general solutions to other indeterminate quadratic and higher-order polynomial equations.


Islamic number theory
From the 9th century, Islamic mathematicians had a keen interest in number theory. The first of these mathematicians was the Arab mathematician Thabit ibn Qurra, who discovered a theorem which allowed pairs of amicable numbers to be found, that is two numbers such that each is the sum of the proper divisors of the other. In the 10th century, Al-Baghdadi looked at a slight variant of Thabit ibn Qurra's theorem.

In the 10th century, al-Haitham seems to have been the first to attempt to classify all even perfect numbers (numbers equal to the sum of their proper divisors) as those of the form 2k − 1(2k − 1) where 2k − 1 is prime. Al-Haytham is also the first person to state Wilson's theorem, namely that if p is prime then 1 + (p − 1)! is divisible by p. It is unclear whether he knew how to prove this result. It is called Wilson's theorem because of a comment made by Edward Waring in 1770 that John Wilson had noticed the result. There is no evidence that John Wilson knew how to prove it and most certainly Waring did not. Lagrange gave the first proof in 1771.

Amicable numbers played a large role in Islamic mathematics. In the 13th century, Persian mathematician Al-Farisi gave a new proof of Thabit ibn Qurra's theorem, introducing important new ideas concerning factorisation and combinatorial methods. He also gave the pair of amicable numbers 17296, 18416 which have been attributed to Euler, but we know that these were known earlier than al-Farisi, perhaps even by Thabit ibn Qurra himself. In the 17th century, Muhammad Baqir Yazdi gave the pair of amicable numbers 9,363,584 and 9,437,056 still many years before Euler's contribution.

Early European number theory
Number theory began in Europe in the 16th and 17th centuries, with François Viète, Bachet de Meziriac, and especially Fermat, whose infinite descent method was the first general proof of diophantine questions. Fermat's last theorem was posed as a problem in 1637, a proof of which wasn't found until 1994. Fermat also posed the equation 61x2 + 1 = y2 as a problem in 1657.

In the eighteenth century, Euler and Lagrange made important contributions to number theory. Euler did some work on analytic number theory, and found a general solution to the equation 61x2 + 1 = y2, which Fermat posed as a problem. Lagrange found a solution to the more general Pell's equation. Euler and Lagrange solved these Pell equations by means of continued fractions, though this was more difficult than the Indian chakravala method.


Beginnings of modern number theory
Around the beginning of the nineteenth century books of Legendre (1798), and Gauss put together the first systematic theories in Europe. Gauss's Disquisitiones Arithmeticae (1801) may be said to begin the modern theory of numbers.

The formulation of the theory of congruences starts with Gauss's Disquisitiones. He introduced the symbolism


and explored most of the field. Chebyshev published in 1847 a work in Russian on the subject, and in France Serret popularised it.

Besides summarizing previous work, Legendre stated the law of quadratic reciprocity. This law, discovered by induction and enunciated by Euler, was first proved by Legendre in his Théorie des Nombres (1798) for special cases. Independently of Euler and Legendre, Gauss discovered the law about 1795, and was the first to give a general proof. To the subject have also contributed: Cauchy; Dirichlet whose Vorlesungen über Zahlentheorie is a classic; Jacobi, who introduced the Jacobi symbol; Liouville, Zeller(?), Eisenstein, Kummer, and Kronecker. The theory extends to include cubic and biquadratic reciprocity, (Gauss, Jacobi who first proved the law of cubic reciprocity, and Kummer).

To Gauss is also due the representation of numbers by binary quadratic forms.

Prime number theory
A recurring and productive theme in number theory is the study of the distribution of prime numbers. Carl Friedrich Gauss conjectured the limit of the number of primes not exceeding a given number (the prime number theorem) as a teenager.

Chebyshev (1850) gave useful bounds for the number of primes between two given limits. Riemann introduced complex analysis into the theory of the Riemann zeta function. This led to a relation between the zeros of the zeta function and the distribution of primes, eventually leading to a proof of prime number theorem independently by Hadamard and de la Vallée Poussin in 1896. However, an elementary proof was given later by Paul Erdős and Atle Selberg in 1949+. Here elementary means that it does not use techniques of complex analysis; however, the proof is still very ingenious and difficult. The Riemann hypothesis, which would give much more accurate information, is still an open question.


Nineteenth-century developments
Cauchy, Poinsot (1845), Lebesgue(?) (1859, 1868), and notably Hermite have added to the subject. In the theory of ternary forms Eisenstein has been a leader, and to him and H. J. S. Smith is also due a noteworthy advance in the theory of forms in general. Smith gave a complete classification of ternary quadratic forms, and extended Gauss's researches concerning real quadratic forms to complex forms. The investigations concerning the representation of numbers by the sum of 4, 5, 6, 7, 8 squares were advanced by Eisenstein and the theory was completed by Smith.

Dirichlet was the first to lecture upon the subject in a German university. Among his contributions is the extension of Fermat's last theorem:


which Euler and Legendre had proven for n = 3,4 (and therefore by implication, all multiples of 3 and 4), Dirichlet showing that . Among the later French writers are Borel; Poincaré, whose memoirs are numerous and valuable; Tannery, and Stieltjes. Among the leading contributors in Germany were Kronecker, Kummer Schering, Bachmann, and Dedekind. In Austria Stolz's Vorlesungen über allgemeine Arithmetik (1885-86), and in England Mathews' Theory of Numbers (Part I, 1892) were scholarly of general works. Genocchi, Sylvester, and J. W. L. Glaisher have also added to the theory.


[edit] Twentieth-century developments
Major figures in twentieth-century number theory include Paul Erdős, Gerd Faltings, G. H. Hardy, Edmund Landau, John Edensor Littlewood, Srinivasa Ramanujan and André Weil.

Milestones in twentieth-century number theory include the proof of Fermat's Last Theorem by Andrew Wiles in 1994 and the proof of the related Taniyama–Shimura theorem in 1999.

Answer 2

Bhaskara charya or Aryabhatta to play games and teach the Royal children.

Sandy

Answer 3

That Lewis M is a fast typer