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Number Theory and Its Applications Conference -- October , , Debrecen, Hungary

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Description Table of Contents. Summary "Addresses contemporary developments in number theory and coding theory, originally presented as lectures at summer school held at Bilkent University, Ankara, Turkey. Includes many results in book form for the first time. Table of Contents Arithmetic progressions of polynomials over a finite field; some function field estimates with applications; topics in analytic number theory; the sieve method; a remark on the nonexistence of generalized bent functions; algebraic independence of pi and e pi; modular forms and Hecke operators; the Mahler measure of polynomials; heights of algebraic points; fibre products, character sums and geometric Goppa codes; Vinogradov's method and some applications; simultaneous approximations and algebraic independence; a survey of results on primes in short intervals.

By the early twentieth century, it had been superseded by "number theory". 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. The first historical find of an arithmetical nature is a fragment of a table: The triples are too many and too large to have been obtained by brute force.

The heading over the first column reads: The table's layout suggests [3] that it was constructed by means of what amounts, in modern language, to the identity. It is not known what these applications may have been, or whether there could have been any; Babylonian astronomy , for example, truly came into its own only later. It has been suggested instead that the table was a source of numerical examples for school problems.

While Babylonian number theory—or what survives of Babylonian mathematics that can be called thus—consists of this single, striking fragment, Babylonian algebra in the secondary-school sense of "algebra" was exceptionally well developed. Much earlier sources [9] state that Thales and Pythagoras traveled and studied in Egypt. The Pythagorean tradition spoke also of so-called polygonal or figurate numbers. We know of no clearly arithmetical material in ancient Egyptian or Vedic sources, though there is some algebra in both.

The Chinese remainder theorem appears as an exercise [16] in Sunzi Suanjing 3rd, 4th or 5th century CE. There is also some numerical mysticism in Chinese mathematics, [note 5] but, unlike that of the Pythagoreans, it seems to have led nowhere. Like the Pythagoreans' perfect numbers, magic squares have passed from superstition into recreation.

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One may also study real numbers in relation to rational numbers, for example, as approximated by the latter Diophantine approximation. The history of each subfield is briefly addressed in its own section below; see the main article of each subfield for fuller treatments. Aside from a few fragments, the mathematics of Classical Greece is known to us either through the reports of contemporary non-mathematicians or through mathematical works from the early Hellenistic period. Euler was generous in giving credit to others Varadarajan , p. The areas below date as such from no earlier than the mid-twentieth century, even if they are based on older material. Much earlier sources [9] state that Thales and Pythagoras traveled and studied in Egypt.

Aside from a few fragments, the mathematics of Classical Greece is known to us either through the reports of contemporary non-mathematicians or through mathematical works from the early Hellenistic period. While Asian mathematics influenced Greek and Hellenistic learning, it seems to be the case that Greek mathematics is also an indigenous tradition.

Number Theory and Its Applications Conference -- October , , Debrecen, Hungary

Eusebius, PE X, chapter 4 mentions of Pythagoras:. Aristotle claimed that the philosophy of Plato closely followed the teachings of the Pythagoreans, [20] and Cicero repeats this claim: Platonem ferunt didicisse Pythagorea omnia "They say Plato learned all things Pythagorean". Plato had a keen interest in mathematics, and distinguished clearly between arithmetic and calculation. By arithmetic he meant, in part, theorising on number, rather than what arithmetic or number theory have come to mean. Theaetetus was, like Plato, a disciple of Theodorus's; he worked on distinguishing different kinds of incommensurables , and was thus arguably a pioneer in the study of number systems.

Euclid devoted part of his Elements to prime numbers and divisibility, topics that belong unambiguously 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 the Euclidean algorithm ; Elements , Prop. In , Lessing published an epigram he had found in a manuscript during his work as a librarian; it claimed to be a letter sent by Archimedes to Eratosthenes. As far as we know, such equations were first successfully treated by the Indian school.

It is not known whether Archimedes himself had a method of solution. Very little is known about Diophantus of Alexandria ; he probably lived in the third century CE, that is, about five hundred years after Euclid. Six out of the thirteen books of Diophantus's Arithmetica survive in the original Greek; four more books survive in an Arabic translation.

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Thus, nowadays, we speak of Diophantine equations when we speak of polynomial equations to which rational or integer solutions must be found. One may say that Diophantus was studying rational points, that is, points whose coordinates are rational—on curves and algebraic varieties ; however, unlike the Greeks of the Classical period, who did what we would now call basic algebra in geometrical terms, Diophantus did what we would now call basic algebraic geometry in purely algebraic terms.

Diophantus also studied the equations of some non-rational curves, for which no rational parametrisation is possible. He managed to find some rational points on these curves elliptic curves , as it happens, in what seems to be their first known occurrence by means of what amounts to a tangent construction: Diophantus also resorted to what could be called a special case of a secant construction.

While Diophantus was concerned largely with rational solutions, he assumed some results on integer numbers, in particular that every integer is the sum of four squares though he never stated as much explicitly. While Greek astronomy probably influenced Indian learning, to the point of introducing trigonometry, [24] it seems to be the case that Indian mathematics is otherwise an indigenous tradition; [25] in particular, there is no evidence that Euclid's Elements reached India before the 18th century.

Brahmagupta CE started the systematic study of indefinite quadratic equations—in particular, the misnamed Pell equation , in which Archimedes may have first been interested, and which did not start to be solved in the West until the time of Fermat and Euler. Later Sanskrit authors would follow, using Brahmagupta's technical terminology. Diophantus's main work, the Arithmetica , was translated into Arabic by Qusta ibn Luqa — Other than a treatise on squares in arithmetic progression by Fibonacci ca. Matters started to change in Europe in the late Renaissance , thanks to a renewed study of the works of Greek antiquity.

A catalyst was the textual emendation and translation into Latin of Diophantus's Arithmetica Bachet , , following a first attempt by Xylander , Pierre de Fermat — never published his writings; in particular, his work on number theory is contained almost entirely in letters to mathematicians and in private marginal notes.

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

The interest of Leonhard Euler — in number theory was first spurred in , when a friend of his, the amateur [note 8] Goldbach , pointed him towards some of Fermat's work on the subject. Joseph-Louis Lagrange — was the first to give full proofs of some of Fermat's and Euler's work and observations—for instance, the four-square theorem and the basic theory of the misnamed "Pell's equation" for which an algorithmic solution was found by Fermat and his contemporaries, and also by Jayadeva and Bhaskara II before them.

Adrien-Marie Legendre — was the first to state the law of quadratic reciprocity. He also conjectured what amounts to the prime number theorem and Dirichlet's theorem on arithmetic progressions. In his Disquisitiones Arithmeticae , Carl Friedrich Gauss — proved the law of quadratic reciprocity and developed the theory of quadratic forms in particular, defining their composition.

He also introduced some basic notation congruences and devoted a section to computational matters, including primality tests. The theory of the division of the circle Algebraic number theory may be said to start with the study of reciprocity and cyclotomy , but truly came into its own with the development of abstract algebra and early ideal theory and valuation theory; see below. A conventional starting point for analytic number theory is Dirichlet's theorem on arithmetic progressions , [70] [71] whose proof introduced L-functions and involved some asymptotic analysis and a limiting process on a real variable.

The use of complex analysis in number theory comes later: The history of each subfield is briefly addressed in its own section below; see the main article of each subfield for fuller treatments. Many of the most interesting questions in each area remain open and are being actively worked on. The term elementary generally denotes a method that does not use complex analysis. Here as elsewhere, an elementary proof may be longer and more difficult for most readers than a non-elementary one.

Number theory has the reputation of being a field many of whose results can be stated to the layperson. At the same time, the proofs of these results are not particularly accessible, in part because the range of tools they use is, if anything, unusually broad within mathematics. Some subjects generally considered to be part of analytic number theory, for example, sieve theory , [note 9] are better covered by the second rather than the first definition: The following are examples of problems in analytic number theory: Some of the most important tools of analytic number theory are the circle method , sieve methods and L-functions or, rather, the study of their properties.

The theory of modular forms and, more generally, automorphic forms also occupies an increasingly central place in the toolbox of analytic number theory. One may ask analytic questions about algebraic numbers , and use analytic means to answer such questions; it is thus that algebraic and analytic number theory intersect.

For example, one may define prime ideals generalizations of prime numbers in the field of algebraic numbers and ask how many prime ideals there are up to a certain size. This question can be answered by means of an examination of Dedekind zeta functions , which are generalizations of the Riemann zeta function , a key analytic object at the roots of the subject. Fields of algebraic numbers are also called algebraic number fields , or shortly number fields.

Algebraic number theory studies algebraic number fields. It could be argued that the simplest kind of number fields viz. For that matter, the 11th-century chakravala method amounts—in modern terms—to an algorithm for finding the units of a real quadratic number field. The grounds of the subject as we know it were set in the late nineteenth century, when ideal numbers , the theory of ideals and valuation theory were developed; these are three complementary ways of dealing with the lack of unique factorisation in algebraic number fields.

Number theory and its applications by Dr. Kotyada Srinivas

The initial impetus for the development of ideal numbers by Kummer seems to have come from the study of higher reciprocity laws, [84] that is, generalisations of quadratic reciprocity. Number fields are often studied as extensions of smaller number fields: For example, the complex numbers C are an extension of the reals R , and the reals R are an extension of the rationals Q. Classifying the possible extensions of a given number field is a difficult and partially open problem. Their classification was the object of the programme of class field theory , which was initiated in the late 19th century partly by Kronecker and Eisenstein and carried out largely in — An example of an active area of research in algebraic number theory is Iwasawa theory.

The Langlands program , one of the main current large-scale research plans in mathematics, is sometimes described as an attempt to generalise class field theory to non-abelian extensions of number fields. The central problem of Diophantine geometry is to determine when a Diophantine equation has solutions, and if it does, how many. The approach taken is to think of the solutions of an equation as a geometric object.

For example, an equation in two variables defines a curve in the plane. More generally, an equation, or system of equations, in two or more variables defines a curve , a surface or some other such object in n -dimensional space. In Diophantine geometry, one asks whether there are any rational points points all of whose coordinates are rationals or integral points points all of whose coordinates are integers on the curve or surface.

If there are any such points, the next step is to ask how many there are and how they are distributed. A basic question in this direction is: What about integer points? An example here may be helpful. This curve happens to be a circle of radius 1 around the origin. The rephrasing of questions on equations in terms of points on curves turns out to be felicitous.

The genus can be defined as follows: Other geometrical notions turn out to be just as crucial. There is also the closely linked area of Diophantine approximations: We are looking for approximations that are good relative to the amount of space that it takes to write the rational: Moreover, several concepts especially that of height turn out to be crucial both in Diophantine geometry and in the study of Diophantine approximations. This question is also of special interest in transcendental number theory: Diophantine geometry should not be confused with the geometry of numbers , which is a collection of graphical methods for answering certain questions in algebraic number theory.

Arithmetic geometry , on the other hand, is a contemporary term for much the same domain as that covered by the term Diophantine geometry. The term arithmetic geometry is arguably used most often when one wishes to emphasise the connections to modern algebraic geometry as in, for instance, Faltings's theorem rather than to techniques in Diophantine approximations.

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The areas below date as such from no earlier than the mid-twentieth century, even if they are based on older material. For example, as is explained below, the matter of algorithms in number theory is very old, in some sense older than the concept of proof; at the same time, the modern study of computability dates only from the s and s, and computational complexity theory from the s. Take a number at random between one and a million. How likely is it to be prime? This is just another way of asking how many primes there are between one and a million. How many divisors will it have altogether, and with what likelihood?

What is the probability that it will have many more or many fewer divisors or prime divisors than the average? Much of probabilistic number theory can be seen as an important special case of the study of variables that are almost, but not quite, mutually independent.

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For example, the event that a random integer between one and a million be divisible by two and the event that it be divisible by three are almost independent, but not quite. If certain algebraic objects say, rational or integer solutions to certain equations can be shown to be in the tail of certain sensibly defined distributions, it follows that there must be few of them; this is a very concrete non-probabilistic statement following from a probabilistic one.

Let A be a set of N integers. These questions are characteristic of arithmetic combinatorics. Its focus on issues of growth and distribution accounts in part for its developing links with ergodic theory , finite group theory , model theory , and other fields. An interesting early case is that of what we now call the Euclidean algorithm.

In its basic form namely, as an algorithm for computing the greatest common divisor it appears as Proposition 2 of Book VII in Elements , together with a proof of correctness. There are two main questions: Anyone can test whether a number is prime or, if it is not, split it into prime factors; doing so rapidly is another matter.

We now know fast algorithms for testing primality , but, in spite of much work both theoretical and practical , no truly fast algorithm for factoring. The difficulty of a computation can be useful: For example, these functions can be such that their inverses can be computed only if certain large integers are factorized.

While many difficult computational problems outside number theory are known, most working encryption protocols nowadays are based on the difficulty of a few number-theoretical problems. On a different note—some things may not be computable at all; in fact, this can be proven in some instances. For instance, in , it was proven, as a solution to Hilbert's 10th problem , that there is no Turing machine which can solve all Diophantine equations.

We would necessarily be speaking of Diophantine equations for which there are no integer solutions, since, given a Diophantine equation with at least one solution, the solution itself provides a proof of the fact that a solution exists. We cannot prove, of course, that a particular Diophantine equation is of this kind, since this would imply that it has no solutions. The number-theorist Leonard Dickson — said "Thank God that number theory is unsullied by any application".

Number Theory and Its Applications

Such a view is no longer applicable to number theory. Moreover number theory is one of the three mathematical subdisciplines rewarded by the Fermat Prize. Robson takes issue with the notion that the scribe who produced Plimpton who had to "work for a living", and would not have belonged to a "leisured middle class" could have been motivated by his own "idle curiosity" in the absence of a "market for new mathematics".

Robson , pp. If we count by threes, there is a remainder 2; if we count by fives, there is a remainder 3; if we count by sevens, there is a remainder 2. Find the number of things. If we count by threes and there is a remainder 2, put down If we count by fives and there is a remainder 3, put down If we count by sevens and there is a remainder 2, put down Add them to obtain and subtract to get the answer.

If we count by threes and there is a remainder 1, put down If we count by fives and there is a remainder 1, put down If we count by sevens and there is a remainder 1, put down When [a number] exceeds , the result is obtained by subtracting If the gestation period is 9 months, determine the sex of the unborn child. Put down 49, add the gestation period and subtract the age. From the remainder take away 1 representing the heaven, 2 the earth, 3 the man, 4 the four seasons, 5 the five phases, 6 the six pitch-pipes, 7 the seven stars [of the Dipper], 8 the eight winds, and 9 the nine divisions [of China under Yu the Great].

If the remainder is odd, [the sex] is male and if the remainder is even, [the sex] is female. Hardy and Wright's book is a comprehensive classic, though its clarity sometimes suffers due to the authors' insistence on elementary methods Apostol n. 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.

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Number Theory and Its Applications Conference -- October , , Debrecen, Hungary

Number theory

Number Theory and Its Applications