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Number theory

(numbertheory)





Traditionally, number theory is that branch of pure mathematics concerned with the properties of integers and containsmany open problems that are easily understood even by non-mathematicians. More generally, the field has come to be concerned witha wider class of problems that arose naturally from the study of integers. Number theory may be subdivided into several fieldsaccording to the methods used and the questions investigated. See for example the list of number theory topics .

The term " arithmetic " is also used to refer to number theory. This is asomewhat older term, which is no longer as popular as it once was. Nevertheless, the term remains prevalent --e.g. in the namesof mathematical fields (arithmetic algebraic geometry and the arithmetic of elliptic curves and surfaces). This sense of the termarithmetic should not be confused with the branch of logic which studies arithmetic inthe sense of formal systems.

In elementary number theory, the integers are studied without use of techniques from other mathematicalfields. Questions of divisibility , the Euclidean algorithm to compute greatest common divisors , factorization of integers into prime numbers , investigation of perfect numbers and congruences belong here. Typical statements are Fermat's little theorem and Euler's theorem extending it, the Chinese remainder theorem and the law of quadratic reciprocity . The properties of multiplicative functions such as the Möbius function and Euler's φfunction are investigated; so are integer sequences such as factorials and Fibonacci numbers .

Many questions in elementary number theory appear simple but may require very deep consideration and new approaches. Examplesare

  • The Goldbach conjecture concerning the expressionof even numbers as sums of two primes,
  • Catalan's conjecture regarding successive integerpowers,
  • The twin prime conjecture about the infinitude of prime pairs , and
  • The Collatz conjecture concerning a simple iteration.

The theory of Diophantine equations has even been shownto be undecidable (see Hilbert's tenthproblem ).


Analytic number theory employs themachinery of calculus and complex analysis to tackle questions about integers. The prime number theorem and the related Riemann hypothesis are examples. Waring'sproblem (representing a given integer as a sum of squares, cubes etc.), the Twin Prime Conjecture (finding infinitely many prime pairs with difference 2) and Goldbach's conjecture (writing even integers as sums of twoprimes) are being attacked with analytical methods as well. Proofs of the transcendence of mathematical constants, such as π or e , are also classified as analytical number theory. While statements about transcendentalnumbers may seem to be removed from the study of integers, they really study the possible values of polynomials with integercoefficients evaluated at, say, e; they are also closely linked to the field of Diophantine approximation , where one investigates "how well" a given real number may beapproximated by a rational one.

In algebraic number theory, the concept of number is expanded to the algebraic numbers which are roots of polynomials with rational coefficients. These domains contain elements analogous to the integers, the so-called algebraic integers . In this setting, the familiar features of theintegers (e.g. unique factorization) need not hold. The virtue of the machinery employed -- Galois theory , field cohomology , class field theory , group representations and L-functions -- is that itallows to recover that order partly for this new class of numbers.

Many number theoretical questions are best attacked by studying them modulo p for all primes p (see finite fields ). This is called localization and it leads to theconstruction of the p-adic numbers ; this field of study is called local analysis and it arises from algebraic number theory.

Geometric number theory (traditionally called geometry of numbers ) incorporates all forms of geometry. It starts with Minkowski's theorem about lattice points in convex sets and investigations of sphere packings .Algebraic geometry, especially the theory of elliptic curves , may alsobe employed. The famous Fermat's last theorem wasproved with these techniques.

Finally, computational numbertheory studies algorithms relevant in number theory. Fast algorithms for prime testing and integerfactorization have important applications in cryptography

History of number theory

The theory of numbers, a favorite study among the ancient Greeks, had its renaissance in the sixteenth andseventeenth centuries in the labors of Viète , Bachet de Meziriac , and especially Fermat . In the eighteenth century Euler and Lagrange contributed to the theory, and at its close the subjectbegan to take scientific form through the great labors of Legendre (1798), and Gauss (1801). With the latter's Disquisitiones Arithmeticae (1801) may be saidto begin the modern theory of numbers.

Chebyshev (1850) gave useful bounds for the number of primes between two givenlimits. Riemann (1859) conjectured the limit of the number of primes not exceeding agiven number (the prime number theorem ), introduced complex analysis into the theoryof the Riemann zeta function , and derived the explicit formulae of prime number theory from its zeroes.

The theory of congruences may be said to start withGauss'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 popularisedit.

Besides summarizing previous work, Legendre stated the law of quadratic reciprocity . This law, discoveredby 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 thesubject have also contributed: Cauchy ; Dirichlet whose Vorlesungen über Zahlentheorie is a classic; Jacobi , who introduced the Jacobisymbol ; Liouville , Zeller (?), Eisenstein , Kummer , and Kronecker . The theory extends to include cubic and biquadraticreciprocity , (Gauss, Jacobi who first proved the law of cubic reciprocity , and Kummer).

To Gauss is also due the representation of numbers by binary quadraticforms . Cauchy, Poinsot (1845), Lebesgue (?) (1859, 1868), and notably Hermite have added to thesubject. 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 completeclassification of ternary quadratic forms, and extended Gauss's researches concerning real quadratic forms to complex forms. Theinvestigations concerning the representation of numbers by the sum of 4, 5, 6, 7, 8 squares were advanced by Eisenstein and thetheory was completed by Smith.

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

xn + yn = zn,

which Euler and Legendre had proved for n = 3,4, 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 are Kronecker,Kummer, Schering , Bachmann , and Dedekind . In Austria Stolz 's Vorlesungen überallgemeine Arithmetik (1885-86), and in England Mathews ' Theory of Numbers (PartI, 1892) are among the most scholarly of general works. Genocchi , Sylvester , and J. W. L. Glaisher have also added to the theory.

Quotations

Mathematics is the queen of the sciences and number theory is the queen of mathematics. Gauss

References

Topics in mathematics related to structure

Abstract algebra | Numbertheory | Algebraic geometry | Group theory | Monoids | Analysis | Topology | Linear algebra | Graph theory | Universal algebra | Category theory | Order theory

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