View source for Number theory

The science of integers (cf. also [[Integer|Integer]]). Integers, together with the simplest geometrical figures, were the first and the most ancient mathematical concepts. Number theory arose from problems in [[Arithmetic|arithmetic]] connected with the multiplication and division of integers.

In Ancient Greece (6th century B.C.) divisibility of integers was studied, and particular subclasses of integers (such as prime numbers, cf. [[Prime number|Prime number]], composite numbers, squares) were distinguished; the structure of perfect numbers (cf. [[Perfect number|Perfect number]]) was studied; and the solution in integers of the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067940/n0679401.png" /> was given, that is, an algorithm was described to construct right-angled triangles with sides of integer lengths. Euclid (3rd century B.C.) in his <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067940/n0679402.png" /> gave a systematic development of the theory of divisibility, based on the so-called [[Euclidean algorithm|Euclidean algorithm]] for finding the [[Greatest common divisor|greatest common divisor]] of two integers, and proved the first theorem in the theory of prime numbers: There are infinitely many prime numbers. Somewhat later, Eratosthenes discovered a method of obtaining prime numbers, which is still called the sieve of Eratosthenes (cf. [[Eratosthenes, sieve of|Eratosthenes, sieve of]]).

A systematization of the problems in number theory, and methods for their solution, was carried out by Diophantus (3rd century) in his Aritmetika, where, in particular, he gave solutions in rational numbers of many algebraic equations of the first and second degrees in several unknowns.

In China, from the 2nd century onwards, in connection with calendar computations, the problem arose of finding the least integer that yields given remainders on division by given numbers. This was solved by Sun-tsi (2nd–6th centuries) and by Tsin Tsiushao (13th century) (cf. [[Chinese remainder theorem|Chinese remainder theorem]]).

In India, Bramagupta (7th century) and Bhaskara (12th century) gave general methods for solving in integers indefinite equations of the first degree in two unknowns, as well as of equations of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067940/n0679403.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067940/n0679404.png" />.

The golden age of number theory in Europe began with the work of P. Fermat (17th century). He investigated the solutions of many equations in integers, and in particular conjectured that the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067940/n0679405.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067940/n0679406.png" />, has no solution in natural numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067940/n0679407.png" /> ([[Fermat great theorem|Fermat great theorem]]); he proved that a prime number of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067940/n0679408.png" /> is a sum of two squares, and asserted one of the basic results of the theory of congruences: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067940/n0679409.png" /> is divisible by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067940/n06794010.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067940/n06794011.png" /> is a prime number ([[Fermat little theorem|Fermat little theorem]]).

L. Euler made an exceptionally important contribution to number theory. He proved Fermat's great theorem for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067940/n06794012.png" />, Fermat's little theorem and a generalization of it, and a whole series of theorems about representations of numbers by binary quadratic forms (cf. [[Binary quadratic form|Binary quadratic form]]). Euler was the first to approach solutions to problems in number theory by means of mathematical analysis, which led to the creation of [[Analytic number theory|analytic number theory]]. Euler's generating functions (cf. [[Generating function|Generating function]]) appeared as the source of the [[Circle method|circle method]] of Hardy–Littlewood–Ramanujan and the method of trigonometric sums (cf. [[Trigonometric sums, method of|Trigonometric sums, method of]]) of I.M. Vinogradov — the basic methods of modern additive number theory, while the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067940/n06794013.png" />, introduced by Euler, and its generalizations, are the basis of the contemporary analytic methods used to investigate the problem of the [[Distribution of prime numbers|distribution of prime numbers]].

In the correspondence between Ch. Goldbach and Euler, three famous problems were posed: Whether every odd <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067940/n06794014.png" /> is a sum of three primes; whether every even <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067940/n06794015.png" /> is a sum of two primes; and whether every odd <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067940/n06794016.png" /> has the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067940/n06794017.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067940/n06794018.png" /> is a prime number and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067940/n06794019.png" /> an integer. The first problem was solved by Vinogradov (1937), and the other two remain unsolved (1989).

C.F. Gauss created the basic methods in and completed the construction of the theory of congruences (cf. [[Congruence|Congruence]]), proved the [[Quadratic reciprocity law|quadratic reciprocity law]] (cf. also [[Gauss reciprocity law|Gauss reciprocity law]]; [[Reciprocity laws|Reciprocity laws]]), which had been formulated by Euler, laid the foundations of the theory of representations of numbers by quadratic forms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067940/n06794020.png" /> and by forms of higher degrees in several variables, introduced the so-called Gauss sums (cf. [[Gauss sum|Gauss sum]])

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067940/n06794021.png" /></td> </tr></table>

which were the first trigonometric sums, and showed that they were useful in solving problems in number theory. While number theory before the time of Gauss had been merely a collection of different results and ideas, after his work it began to develop in several directions as a harmonized theory.

Great contributions to number theory have been made by many scholars in the 19th century and 20th century.

One of the peculiarities and attractions of number theory is the simplicity and accessibility of the majority of its problems, and the difficulty of solving them. For example, the problem of whether there are infinitely many prime [[Twins|twins]] was posed by Euclid and is still unsolved (1989). Individual problems in number theory have become sources of important independent branches of mathematics. Among these are: the theory of prime numbers (cf. [[Prime number|Prime number]]) and the related theory of the [[Zeta-function|zeta-function]] and [[Dirichlet series|Dirichlet series]], the theory of [[Diophantine equations|Diophantine equations]]; [[Additive number theory|additive number theory]], the [[Metric theory of numbers|metric theory of numbers]], the theory of algebraic and transcendental numbers, [[Algebraic number theory|algebraic number theory]], the theory of [[Diophantine approximations|Diophantine approximations]], probabilistic number theory (cf. [[Number theory, probabilistic methods in|Number theory, probabilistic methods in]]), and the [[Geometry of numbers|geometry of numbers]]. For example, a source of analytic number theory was the problem of the distribution of primes in series of natural numbers and the problem of representing natural numbers as sums of terms of a particular form. Solving equations in integers, in particular Fermat's great theorem, was the source of algebraic number theory. The problem of constructing a disc of unit area by means of a straightedge and a compass (cf. [[Quadrature of the circle|Quadrature of the circle]]) led to questions about the arithmetic nature of the number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067940/n06794022.png" />, and hence to the creation of the theory of algebraic and transcendental numbers.

All the above branches of number theory are interconnected, complementing and enriching one another.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> I.M. Vinogradov, "Elements of number theory" , Dover, reprint (1954) (Translated from Russian) {{MR|0062138}} {{ZBL|0057.28201}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> I.M. Vinogradov, "The method of trigonometric sums in the theory of numbers" , Interscience (1954) (Translated from Russian) {{MR|0603100}} {{MR|0409380}} {{ZBL|}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> I.M. Vinogradov, "Basic variants of the method of trigonometric sums" , Moscow (1976) (In Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> A.A. Karatsuba, "Fundamentals of analytic number theory" , Moscow (1983) (In Russian) {{MR|767273}} {{ZBL|}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> Z.I. Borevich, I.R. Shafarevich, "Number theory" , Acad. Press (1966) (Translated from Russian) (German translation: Birkhäuser, 1966) {{MR|0195803}} {{ZBL|0145.04902}} </TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> H. Davenport, "Multiplicative number theory" , Springer (1980) {{MR|0606931}} {{ZBL|0453.10002}} </TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top"> K. Chandrasekharan, "Introduction to analytic number theory" , Springer (1968) {{MR|0249348}} {{ZBL|0169.37502}} </TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top"> H. Hasse, "Zahlentheorie" , Akademie Verlag (1963) {{MR|0153659}} {{ZBL|1038.11500}} </TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top"> P.G.L. Dirichlet, "Vorlesungen über Zahlentheorie" , Vieweg (1894) {{MR|0237283}} {{ZBL|25.0252.01}} </TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top"> E.C. Titchmarsh, "The theory of the Riemann zeta-function" , Clarendon Press (1951) {{MR|0046485}} {{ZBL|0042.07901}} </TD></TR><TR><TD valign="top">[11]</TD> <TD valign="top"> B.A. Venkov, "Elementary number theory" , Wolters-Noordhoff (1970) (Translated from Russian) {{MR|0265267}} {{ZBL|0204.37101}} </TD></TR></table>



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====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> G.H. Hardy, E.M. Wright, "An introduction to the theory of numbers" , Oxford Univ. Press (1979) pp. Chapt. XVI {{MR|0568909}} {{ZBL|0423.10001}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> W.J. Leveque, "Topics in number theory" , '''1''' , Addison-Wesley (1965) {{MR|1942365}} {{MR|1529652}} {{MR|0080682}} {{ZBL|1009.11001}} {{ZBL|0070.03804}} {{ZBL|0070.03803}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> D. Shanks, "Solved and unsolved problems in number theory" , Chelsea, reprint (1978) {{MR|0516658}} {{ZBL|0397.10001}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> A. Weil, "Number theory: an approach through history: from Hammupari to Legendre" , Birkhäuser (1984)</TD></TR></table>