Number theory
La Number theory is the branch of mathematics that studies the properties of the numbers, in particular the integers, but more generally, studies the properties of the rings of numbers: full rings that contain to Z{displaystyle mathbb {Z} } through a finite and injective morphism Z A{displaystyle mathbb {Z} hookrightarrow A}. It contains a considerable amount of problems that could be understood by "not mathematicians". More generally, this field studies the problems that arise with the study of the whole numbers. As quote Jürgen Neukirch:
The theory of numbers occupies among mathematical disciplines an idealized position analogous to that which occupy mathematics itself among the other sciences.
Integers can be considered by themselves or as solutions of equations (Diophantine geometry). Issues in number theory are often best understood through the study of analytic objects (for example, the Riemann zeta function) that encode properties of integers, primes, or other objects of number theory. somehow (Analytic Number Theory). Real numbers can also be studied in relation to rational numbers, for example, as an approximation of the latter (Diophantine approximation).
The term "arithmetic" It was also used to refer to number theory. This is a fairly old term, although not as popular anymore. Hence number theory is often called high arithmetic, although the term has also fallen out of use. This sense of the term arithmetic should not be confused with elementary arithmetic, or with the branch of logic that studies Peano arithmetic as a formal system. Mathematicians who study number theory are called number theorists.
History
Origins
Dawn of Arithmetic
The oldest historical finding of arithmetic character is a fragment of table: the rotated clay tablet Plimpton 322 (Larsa, Mesopotamia, ca. 1800 B.C.) contains a list of "pitagonistic strips", that is, integers (a,b,c){displaystyle (a,b,c)} such that a2+b2=c2{displaystyle a^{2}+b^{2}=c^{2}}. The triples are too large to have been obtained by brute force. The title on the first column says: "The takiltum of the diagonal that has been restored such that the width..."
The arrangement of the table suggests that it was constructed using what is equivalent, in modern language, to the identity
- (12(x− − 1x))2+1=(12(x+1x))2,{displaystyle left({frac {1}{2}}}}left(x-{frac {1}{x}{x}right)^{2}+1=left({frac {1}{2}}}{x+{frac {1}{x}{x}{x}}right)right)^{2}{2}},}
that is implied in the routine exercises of ancient Babylon. If any other method was used, the triples were built first and then reordered by c/a{displaystyle c/a}, presumably for real use as "tabla", for example, with view to applications.
It is not known what these applications may have been, or if there may have been any; Babylonian astronomy, for example, really developed only later. It has been suggested instead that the table was a source of numerical examples for school problems, which is controversial. Robson's article is controversially written in order to "maybe [...] knock [Plimpton 322] off his pedestal";; at the same time, he settles on the conclusion that:
[...] the question "How did the table be calculated?" does not have to have the same answer as the question "what problems does the table pose?" The first can be answered more satisfactorily by reciprocal pairs, as was first suggested half a century ago, and the second by some sort of rectal triangle problems.
Robson takes issue with the idea that the scribe who produced Plimpton 322, who had to "work for a living," and would not have been from an "affluent middle class", may be motivated by his own "idle curiosity" in the absence of a "market for new mathematics".
Whereas Babylonian number theory - or what survives of Babylonian mathematics which may be called that - consists of this single striking fragment, Babylonian algebra (in the secondary sense of "algebra") was exceptionally well developed. Late Neoplatonic sources claim that Pythagoras learned mathematics from the Babylonians. Much earlier sources state that Thales and Pythagoras traveled and studied in Egypt.
Euclides IX 21-34 is very probably pythagorean; it is a very simple material (" pairs are pairs", "if an odd number measures [= divide] a pair number, then also measures [= divide] half of it"), but it is all that is needed to prove that 2{displaystyle {sqrt {2}}}It's an irrational. Pythagoric mystics gave great importance to pairs and odds. The discovery that 2{displaystyle {sqrt {2}}} is irrational is attributed to the first Pythagoreans (preTeodoro). By revealing (in modern terms) that the numbers could be irrational, this discovery seems to have caused the first foundational crisis of the history of mathematics; its demonstration or its disclosure is sometimes attributed to Hipaso, who was expelled or dissected from the Pythagorean sect. This forced the distinction between the numbers (the integers and rationals - the subjects of arithmetic-), on the one hand, and the lengths and proportions (that we would identify with real numbers, whether rational or not), by another.
The Pythagorean tradition also spoke of the so-called polygonal or figurative numbers. While square, cubic, etc. numbers are now seen as more natural than triangular, pentagonal, etc. numbers, the study of sums of triangular and pentagonal numbers would prove fruitful in early modern (17th century to early XIX).
We are not aware of any clearly arithmetic material in the ancient Egyptian or Vedic sources, although there is some algebra in both. The Chinese remainder theorem appears as an exercise in Sunzi Suanjing (3rd, 4th, or 5th century CE). (There is an important step that is overlooked in Sunzi's solution: it is the problem that Kuṭṭaka's Āryabhaṭa later solved - see below.)
There is also some number mysticism in Chinese mathematics, but unlike that of the Pythagoreans, it seems to have led nowhere. Like the perfect numbers of the Pythagoreans, magic squares have passed from superstition to recreation.
Classical Greece and the early Hellenistic period
Apart from a few fragments, classical Greek mathematics is known to us either from the reports of contemporary non-mathematicians or from early Hellenistic mathematical works. In the case of number theory, this means, in general, Plato and Euclid, respectively.
Although Asian mathematics influenced Greek and Hellenistic learning, it appears that Greek mathematics is also an indigenous tradition.
Eusebius of Caesarea, PE X, in chapter 4 mentions Pythagoras:
In fact, said Pythagoras, as he studied the wisdom of each nation, visited Babylon, and Egypt, and all Persia, being instructed by the Magi and the priests: and besides these he realized that he studied with the Brahmans (these are Indian philosophers); and of some he gathered the astrology, of others the geometry, and of others the arithmetic and contrary music, and different things
Aristotle claimed that Plato's philosophy closely followed the teachings of the Pythagoreans, and Cicero repeats this statement: Platonem ferunt didicisse Pythagorea omnia ("They say Plato learned everything Pythagorean& #34;).
Plato had a great interest in mathematics, and clearly distinguished between arithmetic and calculus. (By arithmetic referred, in part, to the theorization of the number, rather than what they have come to mean arithmetic or Number theory). It is through one of Plato's dialogues - namely, the Teteto'- that we know that Teodoro had shown that 3,5,...... ,17{displaystyle {sqrt {3}},{sqrt {5}}},dots{sqrt {17}}}} They're irrational. Teteto was, like Plato, a disciple of Teodoro; he worked on the distinction of the different types of insurable incomes, so it could be said that he was a pioneer in the study of numerical systems. The book X of the Euclid Elements is described by Pappus as largely based on the work of Theaetetus.
Euclid devoted part of his Elements to prime numbers and divisibility, topics that unequivocally belong to number theory and are basic to it (books VII to IX of Euclid's Elements). In particular, he gave an algorithm for computing the greatest common divisor of two numbers (Euclid's algorithm; Elements, Prop. VII.2) and the first known proof of the infinity of prime numbers (Elements, Prop. IX.20).
In 1773, Lessing published an epigram he had found in a manuscript during his work as a librarian; it was purported to be a letter sent by Archimedes to Eratosthenes. The epigram proposed what is known as the Archimedean cattle problem; his solution, absent from the manuscript, requires solving an indeterminate quadratic equation, which reduces to what would later be mistakenly called Pell's equation. As far as we know, such equations were first successfully treated by the Indian school. It is not known if Archimedes himself had a solution method.
Diophantus of Alexandria
It is known very little about Diofanto of Alexandria; it probably lived in the centuryIII of our era, that is, about five hundred years after Euclides. Six of the thirteen books of the Arithmetic of Diofanto are preserved in the original Greek and four more in a translation into Arabic. La Arithmetic is a collection of problems developed in which the task is invariably to find rational solutions to a system of polynomial equations, usually in the form f(x,and)=z2{displaystyle f(x,y)=z^{2}} or f(x,and,z)=w2{displaystyle f(x,y,z)=w^{2}}. Thus, today, we are talking about "Dyphantic events" when we talk about polynomial equations to which we need to find rational or whole solutions.
It can be said that Diofanto studied the rational points, that is, the points whose coordinates are rational, in algebraic curves and varieties; however, unlike the Greeks of the classical period, who did what we would call the basic algebra in geometric terms today, Diofanto did what we would call the basic algebraic geometry in purely algebraic terms. In modern language, what Diofanto did was to find rational parameterizations of the varieties; that is, given an equation of form (say) f(x1,x2,x3)=0{displaystyle f(x_{1},x_{2},x_{3})=0}, its objective was to find (in essence) three rational functions g1,g2,g3{displaystyle g_{1},g_{2},g_{3}}}} such that, for all values of r{displaystyle r} and s{displaystyle s}establish
xi=gi(r,s){displaystyle x_{i}=g_{i}(r,s)} for i=1,2,3{displaystyle i=1,2,3} gives a solution to f(x1,x2,x3)=0.{displaystyle f(x_{1},x_{2},x_{3})=0.}
Diophantus also studied the equations of some non-rational curves, for which no rational parameterization is possible. He managed to find some rational points on these curves (elliptic curve, in what seems to be the first known occurrence of it) by what amounts to a tangent construction: translated into coordinate geometry (which did not exist in Diophantus's time), his method would be visualized as drawing a tangent to a curve at a known rational point, and then finding the other point of intersection of the tangent with the curve; that other point is a new rational point. (Diophantus also resorted to what might be called a special case of the construction of a secant.)
Although Diophantus was largely concerned with rational solutions, he did assume some results about integers, notably that every integer is the sum of four squares, although he never said so explicitly.
Āryabhaṭa, Brahmagupta, Bhāskara
Although Greek astronomy probably influenced Indian learning, to the extent of introducing trigonometry, it seems to be the case that Indian mathematics is otherwise an indigenous tradition; in particular, there is no evidence of that Euclid's Elements reached India before the 18th century.
Àryabhaṭa (476-550 d.C.) showed that pairs of congruences simultaneously nequiva1modm1{displaystyle nequiva_{1}{bmod {m}}{1}}, nequiva2modm2{displaystyle nequiva_{2}{bmod {m}}{2}} could be resolved by means of a method that was called kuṭakaor sprayer; it is a procedure close to (a generalization of) the Euclidean algorithm, which was probably discovered independently in India. Àryabhaṭa seems to have had in mind applications to astronomical calculations.
Brahmagupta (AD 628) began the systematic study of undefined quadratic equations - in particular, the misnamed Pell's Equation, in which Archimedes may have first been interested, and which did not begin to be solved in the West until the epoch of Fermat and Euler. Later Sanskrit authors would follow, using Brahmagupta's technical terminology. A general procedure (the chakravala, or "cyclical method") for solving Pell's equation was eventually found by Jayadeva (cited in the 19th century XI; his work is otherwise lost); the earliest surviving exposition appears in the Bīja-gaṇita of Bhāskara II (12th century).
Indian mathematics remained largely unknown in Europe until the late 18th century; Brahmagupta and Bhāskara's work was translated into English in 1817 by Henry Colebrooke.
Arithmetic in the Islamic Golden Age
At the beginning of the IX century, Caliph Al-Ma'mun ordered the translation of many Greek mathematical works and least one Sanskrit work (the Sindhind, which may or may not be the Brahmagupta of Brāhmasphuṭasiddhānta). Diophantus' main work, Arithmetic, was translated into Arabic by Qusta ibn Luqa (820-912). Part of the treatise al-Fakhri (from al-Karajī, 953 - ca. 1029) is based on it to some extent. According to Rashed Roshdi, Al-Karajī's contemporary Ibn al-Haytham knew what would later be called Wilson's theorem.
Western Europe in the Middle Ages
Apart from a treatise on squares in the Fibonacci arithmetic progression - which he traveled and studied in North Africa and Constantinople - no number theory was done in Western Europe during the Middle Ages. Things began to change in Europe at the end of the Renaissance, thanks to a renewed study of the works of Greek antiquity. One catalyst was the textual emendation and Latin translation of Diophantus's Arithmetica.
Fields
Depending on the methods used and the questions to be answered, number theory is subdivided into various branches.
Elementary number theory
In elementary number theory, whole numbers are studied without employing techniques from other fields of mathematics. Divisibility issues, Euclid's algorithm to calculate the greatest common divisor, the factorization of integers as products of prime numbers, the search for perfect numbers and congruences belong to elementary number theory. Typical statements are Fermat's little theorem and Euler's theorem that extends it, the Chinese remainder theorem, and the law of quadratic reciprocity. In this branch, the properties of multiplicative functions such as the Möbius function and Euler's φ function are investigated; as well as sequences of integers such as factorials and F numbers.
Various questions within elementary number theory seem simple, but require very deep considerations and new approaches, including the following:
- Goldbach guess that all the numbers pair (from 4) are the sum of two prime numbers.
- Conjecture of twin prime numbers on the infinity of so-called twin prime numbers.
- Fermat's last theorem (demonstrated in 1995 by Andrew Wiles).
- Riemann's hypothesis on the distribution of the zeros of the Riemann zeta function, intimately connected with the problem of the distribution of the prime numbers among others.
Analytic Number Theory
An analytic number theory employs calculus and complex analysis as tools to address questions about integers. Some examples of this are the prime number theorem and the Riemann hypothesis. Waring's problem, the twin primes conjecture, and Goldbach's conjecture are also under attack through analytical methods.
Additive Number Theory
Additive number theory deals more deeply with the problems of number representation. Typical problems are those already named, Waring's problem and Goldbach's conjecture. This branch usually uses some results referring to analytic number theory, such as the Hardy-Littlewood circle method, sometimes it is complemented by sieve theory and in some cases topological methods are used.
Algebraic Number Theory
Algebraic number theory is a branch of number theory in which the concept of number is expanded to algebraic numbers, which are the roots of polynomials with rational coefficients.
Geometric Number Theory
Geometric number theory (traditionally called geometry of numbers) incorporates all forms of geometry. It begins with Minkowski's theorem about common points in convex sets and investigations on spherical surfaces.
Combinatorial Number Theory
Combinatorial number theory deals with problems in number theory involving combinatorial ideas and their formulations or solutions. Paul Erdős is the creator of this branch of number theory. Typical topics include covered systems, zero-sum problems, various restricted sets, and arithmetic progressions over a set of integers. Algebraic or analytical methods are quite powerful in this field.
Computational Number Theory
Computational number theory studies the algorithms relevant to number theory. Fast algorithms for evaluating prime numbers and factoring integers have important applications in cryptography.
«The evolution of computing has made arithmetic stop being a contemplative and specialist science to become a true applied branch. The need for new computing algorithms requires - as Enzo R. Gentile says - vast and deep arithmetic knowledge».
History
Mathematicians in India have been interested in finding integer solutions to the Diophantine equations since the mid-1st millennium BCE. The first geometric use of the Diophantine equations dates back to the Shulba-sutras, which were written between the 5th and 3rd centuries B.C. C. The religious Baudhaiana (in the 4th century BC) found two sets of positive integers to a set of simultaneous Diophantine equations, and simultaneous Diophantine equations with more than four unknowns are also used. Apastamba (in the 3rd century BC) used simultaneous Diophantine equations with more than five unknowns.*
The Jain mathematicians were the first to rule out the idea that all infinities are the same or equal, but they had been studied for years. They recognize five different types of infinities: infinite in one or two directions (one-dimensional), infinite on surfaces (two-dimensional), infinite everywhere (three-dimensional), and perpetually infinite (in an infinite number of dimensions).
Number theory was one of the favorite disciplines of study among the Greek mathematicians of Alexandria (in Egypt) from the III century a. C., who were aware of the concept of the Diophantine equation in their particular cases. The first Hellenistic mathematician to study these equations was Diophantus.
Diophantus investigated a method for finding integer solutions to linear indeterminate equations, equations in which sufficient information is missing to produce a unique set of discrete answers. The equation x + y = 5 is an example of such. Diophantus discovered that many indeterminate equations can be reduced to a form where a certain category of solutions are known, even through a solution that is not.
The Diophantine equations were intensively studied by medieval Hindu mathematicians, who were the first to systematically seek methods for the determination of integer solutions. Ariabhata (476-550) gave the first explicit description of the general integer solution of the linear Diophantine equation ay + bx = c , which appears in its text Ariabhatíia. The kuttaka algorithm is considered one of Ariabhata's most significant contributions in pure mathematics, which finds the integer solutions of a system of linear Diophantine equations, a problem of important application in astronomy. He also finds the general solution of the undetermined linear equation using this method.
Brahmagupta (598-668) worked on the most difficult Diophantine equations, which appears in his 18th book dedicated to algebra and indeterminate equations. He used the chakravala method to solve the quadratic Diophantine equations, including those of the form Pell's equation such that 61x 2 + 1 = y2. His Brahma-sphuta-siddhanta was translated into Arabic in 773 and Latin in 1126. The equation 61x2 + 1 = y2 was proposed as a problem by the French mathematician Pierre de Fermat. The general solution of this particular form of Pell's equation was found 70 years later by Leonhard Euler, although the general solution of Pell's equation was found 100 years later by Joseph-Louis de Lagrange in 1767. However, several Centuries earlier, Pell's equation was worked out by Bhaskara II in 1150 using a modified version of Brahmagupta's chakravala method, finding the general solution of other intermediate indeterminate quadratic equations and quadratic Diophantine equations. The chakravala method for finding the general solution of Pell's equation was simpler than the method used by Lagrange 600 years later. Bhaskara also finds the solution of other indeterminate quadratic, cubic, quartic and polynomial equations of higher degrees. Naraian Pandit further refined the other indeterminate quadratics for equations of higher degrees.