Number theory

Traditionally, number theory is that branch of pure mathematics concerned with the properties of integers. It contains many results and open problems that are easily understood, even by non-mathematicians. More generally, the field has come to be concerned with wider classes of problems that have arisen naturally from the study of integers. Number theory may be subdivided into several fields, according to the methods used and the type of questions investigated. See for example the list of number theory topics. Mathematicians working in the field of number theory are called number theorists.

History

Number theory was a favorite study among the Ancient Greeks. It revived in the sixteenth and seventeenth centuries, in Europe, with Viète, Bachet de Meziriac, and especially Fermat. In the eighteenth century Euler and Lagrange made major contributions, and books of Legendre (1798), and Gauss put together the first systematic theories. Gauss's Disquisitiones Arithmeticae (1801) may be said to begin the modern theory of numbers.

Related Topics:
Ancient Greeks - Europe - Viète - Bachet de Meziriac - Fermat - Euler - Lagrange - Legendre - Gauss - Disquisitiones Arithmeticae

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

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:a equiv b pmod c,

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

Related Topics:
Chebyshev - Serret

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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).

Related Topics:
Legendre - Law of quadratic reciprocity - Induction - 1798 - 1795 - Cauchy - Dirichlet - Vorlesungen über Zahlentheorie - Jacobi - Jacobi symbol - Liouville - Zeller - Eisenstein - Kummer - Kronecker - Biquadratic reciprocity - Cubic reciprocity

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

Related Topics:
Quadratic form - Poinsot - 1845 - Lebesgue - 1859 - Hermite - H. J. S. Smith

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

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:x^n+y^n eq z^n,

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which Euler and Legendre had proved for n = 3, 4, Dirichlet showing that x^5+y^5 eq az^5. Among the later French writers are Borel; Poincaré, whose memoirs are numerous and valuable; Tannery, and Stieltjes. Among the leading contributors in Germany are 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) are among the most scholarly of general works. Genocchi, Sylvester, and J. W. L. Glaisher have also added to the theory.

Related Topics:
Borel - Poincaré - Tannery - Stieltjes - Schering - Bachmann - Dedekind - Stolz - Mathews - Genocchi - Sylvester - J. W. L. Glaisher

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A recurring and productive theme in number theory is the study of the distribution of prime numbers. 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 Erd?s 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.

Related Topics:
Gauss - Prime number theorem - Chebyshev - Complex analysis - Riemann zeta function - Hadamard - De la Vallée Poussin - 1896 - Paul Erd?s - Atle Selberg - 1949

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