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We will not ask for credit card numbers or other payment details. J'ai lu et compris le texte ci-dessus. Lien permanent pour contenu public uniquement:. Choisissez le fuseau horaire. Accueil Programme scientifique Ordre du jour Liste des contributions Inscription Liste des participants Contact julien. But a number could be the inverse of itself. When does this happen? I hope this has given you a flavour of what Number Theory is about; there are numerous books available that continue to develop the theory, and large numbers of olympiad problems you might like to tackle with your new knowledge!

Main menu Search. An Introduction to Number Theory. To support this aim, members of the NRICH team work in a wide range of capacities, including providing professional development for teachers wishing to embed rich mathematical tasks into everyday classroom practice. Register for our mailing list.

Number Theory

University of Cambridge. All rights reserved. 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: are there finitely or infinitely many rational points on a given curve or surface?

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.

Number Theory I | Mathematics | MIT OpenCourseWare

Other geometrical notions turn out to be just as crucial. 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 : if a number can be better approximated than any algebraic number, then it is a transcendental number. 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. The areas below date 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. Further: how many prime divisors will it have, on average? 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. Barely larger? 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: "can we compute this? Anyone can test whether a number is prime or, if it is not, split it into prime factors; doing so rapidly is another matter.

A number theory proof

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: modern protocols for encrypting messages for example, RSA depend on functions that are known to all, but whose inverses a are known only to a chosen few, and b would take one too long a time to figure out on one's own. 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.

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.

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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". 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. Answer : Method : 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.

Answer : Male. Method : 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.

Other popular first introductions are:. From Wikipedia, the free encyclopedia. Not to be confused with Numerology. Branch of pure mathematics. Further information: Ancient Greek mathematics.

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Further information: Mathematics in medieval Islam. Main article: Analytic number theory. Main article: Algebraic number theory. Main article: Diophantine geometry. Main article: Probabilistic number theory. Main articles: Arithmetic combinatorics and Additive number theory. Main article: Computational number theory. This section needs expansion with: Modern applications of Number theory. You can help by adding to it. March Heath had to explain: "By arithmetic, Plato meant, not arithmetic in our sense, but the science which considers numbers in themselves, in other words, what we mean by the Theory of Numbers.

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  7. In , Davenport still had to specify that he meant The Higher Arithmetic. Hardy and Wright wrote in the introduction to An Introduction to the Theory of Numbers : "We proposed at one time to change [the title] to An introduction to arithmetic , a more novel and in some ways a more appropriate title; but it was pointed out that this might lead to misunderstandings about the content of the book.

    This is controversial. See Plimpton Robson's article is written polemically Robson , p. This is the last problem in Sunzi's otherwise matter-of-fact treatise. The same was not true in medieval times—whether in the West or the Arab-speaking world—due in part to the importance given to them by the Neopythagorean and hence mystical Nicomachus ca. See van der Waerden , Ch. This notation is actually much later than Fermat's; it first appears in section 1 of Gauss 's Disquisitiones Arithmeticae. Fermat's little theorem is a consequence of the fact that the order of an element of a group divides the order of the group.

    The modern proof would have been within Fermat's means and was indeed given later by Euler , even though the modern concept of a group came long after Fermat or Euler.

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    Weil goes on to say that Fermat would have recognised that Bachet's argument is essentially Euclid's algorithm. There were already some recognisable features of professional practice , viz.