Number theory

Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Mathematics is the queen of the sciences—and number theory is the queen of mathematics."^[1] Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example, rational numbers), or defined as generalizations of the integers (for example, algebraic integers).

For the book by André Weil, see Number Theory: An Approach Through History from Hammurapi to Legendre.

Integers can be considered either in themselves or as solutions to equations (Diophantine geometry). Questions in number theory are often best understood through the study of analytical objects (for example, the Riemann zeta function) that encode properties of the integers, primes or other number-theoretic objects in some fashion (analytic number theory). One may also study real numbers in relation to rational numbers, for example, as approximated by the latter (Diophantine approximation).

The older term for number theory is arithmetic. By the early twentieth century, it had been superseded by "number theory".^{[note 1]} (The word "arithmetic" is used by the general public to mean "elementary calculations"; it has also acquired other meanings in mathematical logic, as in Peano arithmetic, and computer science, as in floating-point arithmetic.) The use of the term arithmetic for number theory regained some ground in the second half of the 20th century, arguably in part due to French influence.^{[note 2]} In particular, arithmetical is commonly preferred as an adjective to number-theoretic.

One of Fermat's first interests was (which appear in Euclid, Elements IX) and amicable numbers;^{[note 6]} these topics led him to work on integer divisors, which were from the beginning among the subjects of the correspondence (1636 onwards) that put him in touch with the mathematical community of the day.^[38]

perfect numbers

In 1638, Fermat claimed, without proof, that all whole numbers can be expressed as the sum of four squares or fewer.

[39]

(1640):^[40] if a is not divisible by a prime p, then $a^{p-1}\equiv 1{\bmod {p}}.$ ^{[note 7]}

Fermat's little theorem

If a and b are , then $a^{2}+b^{2}$ is not divisible by any prime congruent to −1 modulo 4;^[41] and every prime congruent to 1 modulo 4 can be written in the form $a^{2}+b^{2}$ .^[42] These two statements also date from 1640; in 1659, Fermat stated to Huygens that he had proven the latter statement by the method of infinite descent.^[43]

coprime

In 1657, Fermat posed the problem of solving $x^{2}-Ny^{2}=1$ as a challenge to English mathematicians. The problem was solved in a few months by Wallis and Brouncker. Fermat considered their solution valid, but pointed out they had provided an algorithm without a proof (as had Jayadeva and Bhaskara, though Fermat was not aware of this). He stated that a proof could be found by infinite descent.

[44]

Fermat stated and proved (by infinite descent) in the appendix to Observations on Diophantus (Obs. XLV) that $x^{4}+y^{4}=z^{4}$ has no non-trivial solutions in the integers. Fermat also mentioned to his correspondents that $x^{3}+y^{3}=z^{3}$ has no non-trivial solutions, and that this could also be proven by infinite descent.^[46] The first known proof is due to Euler (1753; indeed by infinite descent).^[47]

[45]

Fermat claimed () to have shown there are no solutions to $x^{n}+y^{n}=z^{n}$ for all $n\geq 3$ ; this claim appears in his annotations in the margins of his copy of Diophantus.

Fermat's Last Theorem

Main subdivisions[edit]

Elementary number theory[edit]

The term elementary generally denotes a method that does not use complex analysis. For example, the prime number theorem was first proven using complex analysis in 1896, but an elementary proof was found only in 1949 by Erdős and Selberg.^[77] The term is somewhat ambiguous: for example, proofs based on complex Tauberian theorems (for example, Wiener–Ikehara) are often seen as quite enlightening but not elementary, in spite of using Fourier analysis, rather than complex analysis as such. Here as elsewhere, an elementary proof may be longer and more difficult for most readers than a non-elementary one.

: Public-key encryption schemes such as RSA are based on the difficulty of factoring large composite numbers into their prime factors.^[89]

Cryptography

: The fast Fourier transform (FFT) algorithm, which is used to efficiently compute the discrete Fourier transform, has important applications in signal processing and data analysis.^[90]

Computer science

: The Riemann hypothesis has connections to the distribution of prime numbers and has been studied for its potential implications in physics.^[91]

Physics

: The theory of finite fields and algebraic geometry have been used to construct efficient error-correcting codes.^[92]

Error correction codes

Communications: The design of cellular telephone networks requires knowledge of the theory of , which is a part of analytic number theory.^[93]

modular forms

Study of musical scales: the concept of "", which is the basis for most modern Western music, involves dividing the octave into 12 equal parts.^[94] This has been studied using number theory and in particular the properties of the 12th root of 2.

equal temperament

The number-theorist Leonard Dickson (1874–1954) said "Thank God that number theory is unsullied by any application". Such a view is no longer applicable to number theory.^[86] In 1974, Donald Knuth said "virtually every theorem in elementary number theory arises in a natural, motivated way in connection with the problem of making computers do high-speed numerical calculations".^[87] Elementary number theory is taught in discrete mathematics courses for computer scientists; on the other hand, number theory also has applications to the continuous in numerical analysis.^[88]

Number theory has now several modern applications spanning diverse areas such as:

Prizes[edit]

The American Mathematical Society awards the Cole Prize in Number Theory. Moreover, number theory is one of the three mathematical subdisciplines rewarded by the Fermat Prize.

Algebraic function field

Finite field

p-adic number

List of number theoretic algorithms

This article incorporates material from the article "Number theory", which is licensed under the Creative Commons Attribution-ShareAlike 3.0 Unported License but not under the GFDL.

Citizendium

; E.M. Wright (2008) [1938]. An introduction to the theory of numbers (rev. by D.R. Heath-Brown and J.H. Silverman, 6th ed.). Oxford University Press. ISBN 978-0-19-921986-5. Retrieved 2016-03-02.

G.H. Hardy

(2003) [1954]. Elements of Number Theory (reprint of the 1954 ed.). Mineola, NY: Dover Publications.

Vinogradov, I.M.

Two of the most popular introductions to the subject are:

Hardy and Wright's book is a comprehensive classic, though its clarity sometimes suffers due to the authors' insistence on elementary methods (Apostol 1981). Vinogradov's main attraction consists in its set of problems, which quickly lead to Vinogradov's own research interests; the text itself is very basic and close to minimal. Other popular first introductions are:

Popular choices for a second textbook include:

entry in the Encyclopedia of Mathematics