Number theory

Number theory , or higher arithmetic , is a branch of mathematics that originally studied the properties of integers . In modern number theory, other types of numbers are also considered - for example, algebraic and transcendental , as well as functions of various origin, which are associated with the arithmetic of integers and their generalizations.

In research on number theory, along with arithmetic and algebra , geometric and analytical methods are used, as well as methods of probability theory ^[1] . Methods of number theory are widely used in cryptography , computational mathematics , computer science .

Elementary Number Theory

In elementary number theory, integers are studied without using the methods of other branches of mathematics. Among the main thematic areas of elementary number theory, the following can be distinguished ^[2] .

The theory of divisibility of integers.
Euclidean algorithm for calculating the largest common divisor and the smallest common multiple .
Prime factorization and the main theorem of arithmetic .
The theory of comparisons modulo , the solution of comparisons.
Continued fractions , approximation theory.
Diophantine equations , that is, the solution of indefinite equations in integers.
The study of some classes of integers - perfect numbers , Fibonacci numbers , curly numbers , etc.
Fermat's little theorem and its generalization: Euler's theorem .
Finding Pythagorean triples , the problem of four cubes .
Entertaining math - for example, building magic squares

Analytical Number Theory

In analytical number theory, to derive and prove statements about numbers and numerical functions, a powerful apparatus of mathematical analysis (both real and complex ) is used, sometimes also the theory of differential equations . This allowed to significantly expand the scope of research on number theory. In particular, it included the following new sections ^[2] .

The distribution of primes in a natural number and in other sequences (for example, among the values of a given polynomial).
Representation of natural numbers in the form of sums of terms of a certain type (primes, squares, etc.).
Diophantine approximations .

The first step in applying analytical methods in number theory was the method of generating functions formulated by Euler . To determine the number of integer non-negative solutions of a linear equation of the form

a_{1}x_{1}+...+a_{n}x_{n}=N,

{\ displaystyle a_ {1} x_ {1} + ... + a_ {n} x_ {n} = N,}

Where

a_{1},...,a_{n}

{\ displaystyle a_ {1}, ..., a_ {n}}

- natural numbers ,

Euler constructed a generating function, which is defined as the product of convergent series (for $|z|<1$ ${\ displaystyle | z | <1}$ )

F_{i}(z)=\sum _{k=0}^{\infty }{z_{i}^{a}k}

{\ displaystyle F_ {i} (z) = \ sum _ {k = 0} ^ {\ infty} {z_ {i} ^ {a} k}}

and is the sum of the members of the geometric progression , while

F(z)=\sum _{N=0}^{\infty }l(N)z^{N},

{\ displaystyle F (z) = \ sum _ {N = 0} ^ {\ infty} l (N) z ^ {N},}

Where $l(N)$ ${\ displaystyle l (N)}$ - the number of solutions of the equation being studied. Based on this method, the Hardy – Littlewood circular method was constructed ^[3] .

In his work on the quadratic reciprocity law, Gauss considered finite sums of the form

S(a)=\sum _{n=1}^{p}e^{2\pi ian^{2}/p},

{\ displaystyle S (a) = \ sum _ {n = 1} ^ {p} e ^ {2 \ pi ian ^ {2} / p},}

which can be represented as the sum of sines and cosines (according to the Euler formula ), which is why they are a special case of trigonometric sums ^[3] . The method of trigonometric sums , which allows one to estimate the number of solutions of various equations or systems of equations in integers, plays a large role in analytical number theory. The basis of the method was developed and first applied to problems of number theory by I. M. Vinogradov .

Working on the proof of the Euclidean theorem on the infinity of primes, Euler considered the product of all primes and formulated the identity:

\Pi _{p}\left(1-{\frac {1}{p^{s}}}\right)^{-1}=\sum _{n=1}^{\infty }{\frac {1}{n^{s}}},

{\ displaystyle \ Pi _ {p} \ left (1 - {\ frac {1} {p ^ {s}}} \ right) ^ {- 1} = \ sum _ {n = 1} ^ {\ infty} {\ frac {1} {n ^ {s}}},}

which became the basis for theories of zeta functions ^[3] . The most famous and still unsolved problem of analytic number theory is the proof of the Riemann hypothesis on the zeros of the zeta function , which states that all non-trivial roots of the equation $\zeta (s)=0$ ${\ displaystyle \ zeta (s) = 0}$ lie on the so-called critical line $\mathrm {Re} \,s={\frac {1}{2}}$ ${\ displaystyle \ mathrm {Re} \, s = {\ frac {1} {2}}}$ where $\zeta (s)$ ${\ displaystyle \ zeta (s)}$ - Riemann zeta function .

To prove the theorem on the infinity of primes in a general form, Dirichlet used products on all primes similar to the Euler product and showed that

\Pi _{p}\left(1-{\frac {\chi (p)}{p^{s}}}\right)^{-1}=\sum _{n=1}^{\infty }{\frac {\chi (n)}{n^{s}}}

{\ displaystyle \ Pi _ {p} \ left (1 - {\ frac {\ chi (p)} {p ^ {s}}} \ right) ^ {- 1} = \ sum _ {n = 1} ^ {\ infty} {\ frac {\ chi (n)} {n ^ {s}}}}

,

while the function $\chi (p)$ ${\ displaystyle \ chi (p)}$ The Dirichlet character, called the character, is defined in such a way that it satisfies the following conditions: it is periodic, completely multiplicative and not identically equal to zero. The Dirichlet characters and series have found application in other branches of mathematics, in particular, in algebra , topology, and function theory ^[3] .

Chebyshev showed that the number of primes not exceeding $X$ ${\ displaystyle X}$ designated as $\pi (X)$ ${\ displaystyle \ pi (X)}$ tends to infinity according to the following law ^[3] :

a{\frac {X}{\ln(X)}}<\pi (X)<b{\frac {X}{\ln(X)}},

{\ displaystyle a {\ frac {X} {\ ln (X)}} <\ pi (X) <b {\ frac {X} {\ ln (X)}},}

Where $a>1/2\ln 2$ ${\ displaystyle a> 1/2 \ ln 2}$ and $b<2\ln 2$ ${\ displaystyle b <2 \ ln 2}$ .

The use of complex analysis has greatly enhanced our knowledge of the distribution of primes .

Algebraic Number Theory

In algebraic number theory, the concept of an integer expands; the roots of polynomials with rational coefficients are considered as algebraic numbers. A general theory of algebraic and transcendental numbers was developed. In this case, the algebraic integers , that is, the roots of unitary polynomials with integer coefficients, are the analogs of integers. Unlike integers, the factoriality property, that is, the uniqueness of factorization, is not necessarily fulfilled in the ring of integer algebraic numbers.

The theory of algebraic numbers owes its appearance to the study of Diophantine equations , including attempts to prove Fermat's theorem . Kummer owns equality

x^{n}=z^{n}-y^{n}=\prod _{i=1}^{n}(z-a_{i}y),

{\ displaystyle x ^ {n} = z ^ {n} -y ^ {n} = \ prod _ {i = 1} ^ {n} (z-a_ {i} y),}

Where $a_{i}$ ${\ displaystyle a_ {i}}$ - degree roots $n$ ${\ displaystyle n}$ out of one. Thus, Kummer defined new integers of the form $z+a_{i}y$ ${\ displaystyle z + a_ {i} y}$ . Liouville later showed that if an algebraic number is the root of an equation of degree $n$ ${\ displaystyle n}$ , then you can’t get closer than $Q^{-n}$ ${\ displaystyle Q ^ {- n}}$ approaching fractions of the form $P/Q$ ${\ displaystyle P / Q}$ where $P$ ${\ displaystyle P}$ and $Q$ ${\ displaystyle Q}$ - integer mutually prime numbers ^[3] .

After determining algebraic and transcendental numbers in algebraic number theory, a direction has been identified that deals with proving the transcendence of specific numbers, and a direction that deals with algebraic numbers and studies the degree of their approximation by rational and algebraic ^[3] .

Algebraic number theory includes such sections as divisor theory, Galois theory, class field theory, Dirichlet zeta and L -functions , group cohomology, and much more.

One of the main techniques is to embed the algebraic number field in its completion by some of the metrics - Archimedean (for example, in the field of real or complex numbers) or non-Archimedean (for example, in the field of p -adic numbers ).

Historical Review

Number Theory in the Ancient World

Nameplate Plympton 322

In ancient Egypt, mathematical operations were carried out on integers and aliquots ^[4] . Mathematical papyruses contain problems with solutions and auxiliary tables ^[5] . An even wider use of tables is characteristic of Babylon, which, following the Sumerians, used a six-decimal number system. Babylonian cuneiform mathematical texts include multiplication tables and inverse numbers, squares and cubes of natural numbers ^[6] . In Babylon, there were many Pythagorean triples, for the search of which they probably used an unknown general technique ^[7] . The oldest archaeological find in the history of arithmetic is a fragment of the clay tablet Plympton, 322 , dating from 1800 BC. It contains a list of Pythagorean triples , that is, natural numbers $(a,b,c)$ ${\ displaystyle (a, b, c)}$ such that $a^{2}+b^{2}=c^{2}$ ${\ displaystyle a ^ {2} + b ^ {2} = c ^ {2}}$ . Five-digit numbers are found in triples, and there are too many of them to assume that they were obtained by mechanical enumeration of options ^[1] .

The Pythagoreans, Euclid and Diophantus made a significant contribution to the formation of number theory. The Pythagoreans considered only positive integers and considered the number as a collection of units. The units were indivisible and arranged in the form of regular geometric bodies. The Pythagoreans are characterized by the definition of " curly numbers " ("triangular", "square" and others). Studying the properties of numbers, they broke them into even and odd, simple and compound. Probably, it was the Pythagoreans who, using only the sign of divisibility into two, were able to prove that if $1+2+...+2^{n}=p$ ${\ displaystyle 1 + 2 + ... + 2 ^ {n} = p}$ Is a prime then $2^{n}p$ ${\ displaystyle 2 ^ {n} p}$ Is a perfect number . The proof is presented in the "Beginnings" of Euclid (IX, 36). only in the XVIII century Euler proved that there are no other even perfect numbers, and the question of the infinity of the number of perfect numbers has not yet been resolved. The Pythagoreans also found an infinite number of integer solutions of the equation $x^{2}+y^{2}=z^{2}$ ${\ displaystyle x ^ {2} + y ^ {2} = z ^ {2}}$ , the so-called Pythagorean triples, and derived a general formula for them ^[8] .

The theory of divisibility appeared in 399 BC. e. and apparently belongs to Teetet . Euclid dedicated to her the book VII “Beginnings” and part of book IX. The theory is based on the Euclidean algorithm for finding the common greatest divisor of two numbers. A consequence of the algorithm is the possibility of decomposing any number into prime factors, as well as the uniqueness of such a decomposition. The law of uniqueness of factorization is the basis of the arithmetic of integers ^[9] .

The VII, VIII and IX books included in the Principles of Euclid are devoted to prime numbers and divisibility . In particular, it describes an algorithm for finding the greatest common divisor of two numbers (Euclidean algorithm) and proves the infinity of the set of primes ^[10] .

Diophantus of Alexandria , unlike previous mathematicians of ancient Greece , solved the problems of classical algebra, describing them geometrically. In his work “Arithmetic,” he lists the problems of finding integer solutions for systems of polynomial equations (now called Diophantine equations ) ^[10] . Diophantus's work on solving indefinite equations in rational numbers is at the junction of number theory and algebraic geometry. He explores a second-order equation in two variables $F_{2}(x,y)=0$ ${\ displaystyle F_ {2} (x, y) = 0}$ which is the equation of the conical section . The method by which Diophantus finds rational points of a curve, if at least one of them is known, establishes that a second-order curve either contains an infinite number of points whose coordinates are expressed as rational functions of one parameter, or does not contain them at all. To study equations of the third and fourth order, more complex geometric methods are used (constructing a tangent at a rational point, or a line through two rational points to search for the next intersection) ^[11] .

Number Theory in the Middle Ages

The Chinese remainder theorem was included as an exercise in Sun Tzu 's treatise “Sun Tzu Suan Jing” ( Chinese ex. 孙子算经 , pinyin : sūnzǐ suànjīng ) ^[10] . In his decision one of the important steps was omitted, the full proof was first obtained by Ariabhata in the VI century A.D. e. .

Indian mathematicians Ariabhat, Brahmagupta and Bhaskara solved Diophantine equations of the form $ax+b=cy$ ${\ displaystyle ax + b = cy}$ in integers. In addition, they solved in integers equations of the form $ax^{2}+b=y^{2}$ ${\ displaystyle ax ^ {2} + b = y ^ {2}}$ ^[10] , which was the highest achievement of Indian mathematicians in the field of number theory. Subsequently, this equation and its special case for $b=1$ ${\ displaystyle b = 1}$ attracted the attention of Fermat, Euler, Lagrange. The method proposed by Lagrange to find a solution was close to Indian ^[12] .

Further Development of Number Theory

The theory of numbers was further developed in Fermat's works related to the solution of Diophantine equations and the divisibility of integers. In particular, Fermat formulated a theorem that for any prime $p$ ${\ displaystyle p}$ and the whole $a$ ${\ displaystyle a}$ , $a^{p}-a$ ${\ displaystyle a ^ {p} -a}$ divided by $p$ ${\ displaystyle p}$ , called Fermat’s small theorem and, in addition, formulated a theorem on the unsolvability of the Diophantine equation in integers, or Fermat’s Great Theorem ^[13] . Euler ^[14] was engaged in the generalization of the small theorem and the proof of the great theorem for special cases at the beginning of the 18th century. He began to use the powerful apparatus of mathematical analysis to solve problems in number theory, formulating the method of generating functions, Euler's identity , and also problems associated with addition of primes ^[3] .

In the 19th century, many prominent scientists worked on number theory. Gauss created the theory of comparisons, with the help of which he proved a number of prime theorems, studied the properties of quadratic residues and non-residues, including the quadratic reciprocity law ^[14] , in search of the proof of which Gauss considered finite series of a certain kind, generalized subsequently to trigonometric sums. Developing the work of Euler, Gauss and Dirichlet created the theory of quadratic forms. In addition, they formulated a number of problems on the number of integer points in regions on the plane, particular solutions of which made it possible to prove the general theorem on the infinity of the number of simple points in progressions of the form $nk+l$ ${\ displaystyle nk + l}$ where $k$ ${\ displaystyle k}$ and $l$ ${\ displaystyle l}$ are mutually simple ^[14] . Further study of the distribution of primes was carried out by Chebyshev ^[15] , who showed a more accurate law of the tendency to infinity of primes than Euclid’s theorem, proved Bertrand’s conjecture on the existence of a prime $(x,2x),x\geq 2$ ${\ displaystyle (x, 2x), x \ geq 2}$ , and also posed the problem of estimating from above the smallest value of the difference between adjacent primes (expanding the question of prime twins) ^[3] .

At the beginning of the 20th century, A. N. Korkin , E. I. Zolotarev, and A. A. Markov continued to work on the theory of quadratic forms. Korkin and Zolotarev proved the theorem on variables of positive quaternary quadratic form, and Markov studied the minima of binary quadratic forms of the positive determinant. The formulas formulated by Dirichlet for integer points in areas on the plane were developed in the works of G.F. Voronoi, who in 1903 determined the order of the remainder term. In 1906, the method was successfully transferred to the Gauss problem on the number of integer points in a circle by V. Sierpinski ^[3] .

In 1909, D. Hilbert solved the Waring additive problem ^[3] .

E. Kummer, trying to prove Fermat's theorem, worked with an algebraic number field, for the set of numbers of which he applied all four algebraic operations and thus constructed the arithmetic of the integers of the algebraic number field generated by $a_{i}$ ${\ displaystyle a_ {i}}$ , introduced the concept of ideal factors and gave impetus to the creation of algebraic number theory. In 1844, J. Liouville introduced the concepts of algebraic and transcendental numbers , thus formulating in mathematical terms Euler's remark that the square roots and logarithms of integers have fundamental differences. Liouville showed that algebraic numbers are poorly approximated by rational fractions. At the end of the 19th century, mathematicians such as Charles Hermite , who in 1873 proved the transcendence of numbers, worked to prove the transcendence of specific numbers. $e$ ${\ displaystyle e}$ , F. Lindeman , who in 1882 proved the transcendence of the number $\pi$ ${\ displaystyle \ pi}$ . Another direction was the study of the degree of approximation of algebraic numbers by rational or algebraic ones. Axel Thue worked in it, who in 1909 proved a theorem named after him ^[3] .

Another area of work was Riemann's definition of the zeta function and the proof that it analytically extends to the entire plane of the complex variable and has a number of other properties. Riemann also conjectured the zeros of the zeta function. Работая над дзета-функциями, Ш. ла Валле Пуссен и Жак Адамар сформулировали в 1896 году асимптотический закон распределения простых чисел. Использованный ими метод получения асимптотических формул, или метод комплексного интегрирования, стал широко использоваться в дальнейшем ^[3] .

В первой половине XX века над проблемами теории чисел работали Герман Вейль , сформулировавший соотношение для равномерного распределения дробных долей целочисленных функций, Г.Харди и Дж. Литлвуд, которые сформулировали круговой метод решения аддитивных задач, А. О. Гельфонд и Т. Гнейдер, которые решили 7-ю проблему Гильберта , К. Зигель , который доказал ряд теорем о трансцендентности значений функций, Б. Н. Делоне и Д. К. Фаддеев , которые занимались исследованием диофантова уравнения $x^{3}-ay^{3}=1$ $x^{3}-ay^{3}=1$ , А.Сельберг , который работал в теории дзета-функции Римана ^[3] .

Большой вклад в развитие теории чисел внёс И. М. Виноградов, доказавший неравенство о числе квадратичных вычетов и невычетов на отрезке, определивший метод тригонометрических сумм, который позволил упростить решение проблемы Варинга , а также решение ряда задач по распределению дробных долей функции, определению целых точек в области на плоскости и в пространстве, порядок роста дзета-функции в критической полосе. В задачах, связанных с тригонометрическими суммами, важным является как можно более точная оценка их модуля. Виноградов предложил два метода такой оценки. Кроме того, он вместе с учениками разработал ряд методов, которые позволяют решить задачи, выводимые из гипотезы Римана ^[3] .

Многочисленные работы по теории чисел относятся ко второй половине XX века. Ю. В. Линник разработал дисперсионный метод, который позволил вывести асимптотические формулы для проблемы Харди — Литлвуда и проблемы простых делителей Титчмарша ^[3] .

Вместе с тем, в теории чисел существует большое количество открытых проблем .

Notes

↑ ¹ ² Number Theory, page 1 (англ.) . Encyclopædia Britannica . Дата обращения 6 июня 2012. Архивировано 22 июня 2012 года.
↑ ¹ ² Нестеренко Ю. В., 2008 , с. 3—6.
↑ ¹ ² ³ ⁴ ⁵ ⁶ ⁷ ⁸ ⁹ ¹⁰ ¹¹ ¹² ¹³ ¹⁴ ¹⁵ ¹⁶ Чисел теория // Большая советская энциклопедия
↑ История математики, том I, 1970 , с. 9.
↑ Арифметика // Большая советская энциклопедия
↑ История математики, том I, 1970 , с. 37-39.
↑ История математики, том I, 1970 , с. 50.
↑ История математики, том I, 1970 , с. 68-69.
↑ История математики, том I, 1970 , с. 74-76.
↑ ¹ ² ³ ⁴ Number Theory, page 2 (англ.) . Encyclopædia Britannica. Дата обращения 6 июня 2012. Архивировано 22 июня 2012 года.
↑ History of Mathematics, Volume I, 1970 , p. 146-148.
↑ History of Mathematics, Volume I, 1970 , p. 194-195.
↑ Number Theory, page 3 . Encyclopædia Britannica. Date of treatment June 6, 2012. Archived June 22, 2012.
↑ ¹ ² ³ Number Theory, page 4 . Encyclopædia Britannica. Date of treatment June 6, 2012. Archived June 22, 2012.
↑ Number Theory, page 5 . Encyclopædia Britannica. Date of treatment June 6, 2012. Archived June 22, 2012.

Literature

Aierland K., Rosen M. A classic introduction to modern number theory = A Classical Introduction to Modern Number Theory. - M .: World, 1987.
Borevich Z. I., Shafarevich I. R. Theory of numbers . - M .: Nauka, 1972.- 510 p. Archived January 8, 2011 on Wayback Machine
Vinogradov I. M. Fundamentals of number theory . - M.-L .: GITTL, 1952. - 180 p.
The history of mathematics. From ancient times to the beginning of the New Time // History of Mathematics / Edited by A. Yushkevich , in three volumes. - M .: Nauka, 1970 .-- T. I.
Koch H. Algebraic number theory . - M .: VINITI , 1990.- T. 62.- 301 s. - (The results of science and technology. Series "Modern problems of mathematics. Fundamental directions.").
Manin Yu. I., Panchishkin A. A. Introduction to number theory . - M .: VINITI , 1990.- T. 49.- 341 p. - (The results of science and technology. Series "Modern problems of mathematics. Fundamental directions.").
Nesterenko Yu. V. Number theory: a textbook for students. higher textbook. institutions. - M .: Publishing Center "Academy", 2008. - 272 p. - ISBN 978-5-7695-4646-4 .
Sizyi S.V. Lectures on number theory . - Yekaterinburg: Ural State University named after A.M. Gorky, 1999.
Khinchin A. Ya. Three pearls of number theory . - M .: Nauka, 1979.- 64 p.

Links

Number theory video lecture

[brit-num-theory1-1] ¹ ² Number Theory, page 1 (англ.) . Encyclopædia Britannica . Дата обращения 6 июня 2012. Архивировано 22 июня 2012 года.

[NEST3-2] ¹ ² Нестеренко Ю. В., 2008 , с. 3—6.

[bse-3] ¹ ² ³ ⁴ ⁵ ⁶ ⁷ ⁸ ⁹ ¹⁰ ¹¹ ¹² ¹³ ¹⁴ ¹⁵ ¹⁶ Чисел теория // Большая советская энциклопедия

[_50ead558465485aa-4] История математики, том I, 1970 , с. 9.

[bse_arithmetic-5] Арифметика // Большая советская энциклопедия

[_a9c3de09265e3418-6] История математики, том I, 1970 , с. 37-39.

[_d38e2aff819f1922-7] История математики, том I, 1970 , с. 50.

[_5cea483328d29b33-8] История математики, том I, 1970 , с. 68-69.

[_ca51f72a492c074e-9] История математики, том I, 1970 , с. 74-76.

[brit-num-theory2-10] ¹ ² ³ ⁴ Number Theory, page 2 (англ.) . Encyclopædia Britannica. Дата обращения 6 июня 2012. Архивировано 22 июня 2012 года.

[_70764c8606398dca-11] History of Mathematics, Volume I, 1970 , p. 146-148.

[_b8b029df0c9f5245-12] History of Mathematics, Volume I, 1970 , p. 194-195.

[brit-num-theory3-13] Number Theory, page 3 . Encyclopædia Britannica. Date of treatment June 6, 2012. Archived June 22, 2012.

[brit-num-theory4-14] ¹ ² ³ Number Theory, page 4 . Encyclopædia Britannica. Date of treatment June 6, 2012. Archived June 22, 2012.

[brit-num-theory5-15] Number Theory, page 5 . Encyclopædia Britannica. Date of treatment June 6, 2012. Archived June 22, 2012.