\n* Modern applications of Number theory"},"date":{"wt":"March 2016"}},"i":0}}]}" id="mwAzc">
The number-theorist Leonard Dickson (1874–1954) said "Thank God that number theory is unsullied by any application". Such a view is no longer applicable to number theory.^{ [85] } In 1974, Donald Knuth said "...virtually every theorem in elementary number theory arises in a natural, motivated way in connection with the problem of making computers do high-speed numerical calculations".^{ [86] } Elementary number theory is taught in discrete mathematics courses for computer scientists; on the other hand, number theory also has applications to the continuous in numerical analysis.^{ [87] } As well as the well-known applications to cryptography, there are also applications to many other areas of mathematics.^{ [88] }^{ [89] }^{[ specify ]}
The American Mathematical Society awards the Cole Prize in Number Theory . Moreover number theory is one of the three mathematical subdisciplines rewarded by the Fermat Prize .
[...] the question "how was the tablet calculated?" does not have to have the same answer as the question "what problems does the tablet set?" The first can be answered most satisfactorily by reciprocal pairs, as first suggested half a century ago, and the second by some sort of right-triangle problems ( Robson 2001 , p. 202).Robson takes issue with the notion that the scribe who produced Plimpton 322 (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 2001 , pp. 199–200)
[26] Now there are an unknown number of things. 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: 23.
Method: If we count by threes and there is a remainder 2, put down 140. If we count by fives and there is a remainder 3, put down 63. If we count by sevens and there is a remainder 2, put down 30. Add them to obtain 233 and subtract 210 to get the answer. If we count by threes and there is a remainder 1, put down 70. If we count by fives and there is a remainder 1, put down 21. If we count by sevens and there is a remainder 1, put down 15. When [a number] exceeds 106, the result is obtained by subtracting 105.
[36] Now there is a pregnant woman whose age is 29. If the gestation period is 9 months, determine the sex of the unborn child. Answer: Male.This is the last problem in Sunzi's otherwise matter-of-fact treatise.
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.
In mathematics, a Diophantine equation is a polynomial equation, usually involving two or more unknowns, such that the only solutions of interest are the integer ones. A linear Diophantine equation equates to a constant the sum of two or more monomials, each of degree one. An exponential Diophantine equation is one in which unknowns can appear in exponents.
Diophantus of Alexandria was an Alexandrian mathematician, who was the author of a series of books called Arithmetica, many of which are now lost. His texts deal with solving algebraic equations. Diophantine equations and of Diophantine approximations are important areas of mathematical research. Diophantus coined the term παρισότης (parisotes) to refer to an approximate equality. This term was rendered as adaequalitas in Latin, and became the technique of adequality developed by Pierre de Fermat to find maxima for functions and tangent lines to curves. Diophantus was the first Greek mathematician who recognized fractions as numbers; thus he allowed positive rational numbers for the coefficients and solutions. In modern use, Diophantine equations are usually algebraic equations with integer coefficients, for which integer solutions are sought.
Pell's equation, also called the Pell–Fermat equation, is any Diophantine equation of the form where n is a given positive nonsquare integer and integer solutions are sought for x and y. In Cartesian coordinates, the equation is represented by a hyperbola; solutions occur wherever the curve passes through a point whose x and y coordinates are both integers, such as the trivial solution with x = 1 and y = 0. Joseph Louis Lagrange proved that, as long as n is not a perfect square, Pell's equation has infinitely many distinct integer solutions. These solutions may be used to accurately approximate the square root of n by rational numbers of the form x/y.
Fermat's little theorem states that if p is a prime number, then for any integer a, the number a^{p} − a is an integer multiple of p. In the notation of modular arithmetic, this is expressed as
Hilbert's tenth problem is the tenth on the list of mathematical problems that the German mathematician David Hilbert posed in 1900. It is the challenge to provide a general algorithm which, for any given Diophantine equation, can decide whether the equation has a solution with all unknowns taking integer values.
The modularity theorem states that elliptic curves over the field of rational numbers are related to modular forms. Andrew Wiles proved the modularity theorem for semistable elliptic curves, which was enough to imply Fermat's Last Theorem. Later, a series of papers by Wiles's former students Brian Conrad, Fred Diamond and Richard Taylor, culminating in a joint paper with Christophe Breuil, extended Wiles's techniques to prove the full modularity theorem in 2001.
Algebraic number theory is a branch of number theory that uses the techniques of abstract algebra to study the integers, rational numbers, and their generalizations. Number-theoretic questions are expressed in terms of properties of algebraic objects such as algebraic number fields and their rings of integers, finite fields, and function fields. These properties, such as whether a ring admits unique factorization, the behavior of ideals, and the Galois groups of fields, can resolve questions of primary importance in number theory, like the existence of solutions to Diophantine equations.
In number theory, the study of Diophantine approximation deals with the approximation of real numbers by rational numbers. It is named after Diophantus of Alexandria.
In mathematics, a proof by infinite descent, also known as Fermat's method of descent, is a particular kind of proof by contradiction used to show that a statement cannot possibly hold for any number, by showing that if the statement were to hold for a number, then the same would be true for a smaller number, leading to an infinite descent and ultimately a contradiction. It is a method which relies on the well-ordering principle, and is often used to show that a given equation, such as a Diophantine equation, has no solutions.
In mathematics, Diophantine geometry is the study of points of algebraic varieties with coordinates in the integers, rational numbers, and their generalizations. These generalizations typically are fields that are not algebraically closed, such as number fields, finite fields, function fields, and p-adic fields. It is a sub-branch of arithmetic geometry and is one approach to the theory of Diophantine equations, formulating questions about such equations in terms of algebraic geometry.
In mathematics, Ribet's theorem is a statement in number theory concerning properties of Galois representations associated with modular forms. It was proposed by Jean-Pierre Serre and proven by Ken Ribet. The proof of the epsilon conjecture was a significant step towards the proof of Fermat's Last Theorem. As shown by Serre and Ribet, the Taniyama–Shimura conjecture and the epsilon conjecture together imply that Fermat's Last Theorem is true.
In mathematics, arithmetic geometry is roughly the application of techniques from algebraic geometry to problems in number theory. Arithmetic geometry is centered around Diophantine geometry, the study of rational points of algebraic varieties.
In mathematics, a congruent number is a positive integer that is the area of a right triangle with three rational number sides. A more general definition includes all positive rational numbers with this property.
This is a glossary of arithmetic and diophantine geometry in mathematics, areas growing out of the traditional study of Diophantine equations to encompass large parts of number theory and algebraic geometry. Much of the theory is in the form of proposed conjectures, which can be related at various levels of generality.
In number theory, a congruum is the difference between successive square numbers in an arithmetic progression of three squares. That is, if x^{2}, y^{2}, and z^{2} are three square numbers that are equally spaced apart from each other, then the spacing between them, z^{2} − y^{2} = y^{2} − x^{2}, is called a congruum.
A modular elliptic curve is an elliptic curve E that admits a parametrisation X_{0}(N) → E by a modular curve. This is not the same as a modular curve that happens to be an elliptic curve, something that could be called an elliptic modular curve. The modularity theorem, also known as the Taniyama–Shimura conjecture, asserts that every elliptic curve defined over the rational numbers is modular.
Algebra is one of the broad areas of mathematics, together with number theory, geometry and analysis. In its most general form, algebra is the study of mathematical symbols and the rules for manipulating these symbols; it is a unifying thread of almost all of mathematics. It includes everything from elementary equation solving to the study of abstractions such as groups, rings, and fields. The more basic parts of algebra are called elementary algebra; the more abstract parts are called abstract algebra or modern algebra. Elementary algebra is generally considered to be essential for any study of mathematics, science, or engineering, as well as such applications as medicine and economics. Abstract algebra is a major area in advanced mathematics, studied primarily by professional mathematicians.
In number theory, Fermat's Last Theorem states that no three positive integers a, b, and c satisfy the equation a^{n} + b^{n} = c^{n} for any integer value of n greater than 2. The cases n = 1 and n = 2 have been known since antiquity to have infinitely many solutions.
Fermat's right triangle theorem is a non-existence proof in number theory, published in 1670 among the works of Pierre de Fermat, soon after his death. It is the only complete proof given by Fermat. It has several equivalent formulations, one of which was stated in 1225 by Fibonacci. In its geometric forms, it states:
In mathematics, a diophantine m-tuple is a set of m positive integers such that is a perfect square for any . A set of m positive rational numbers with the similar property that the product of any two is one less than a rational square is known as a rational diophantine m-tuple.
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has extra text (help)Two of the most popular introductions to the subject are:
Hardy and Wright's book is a comprehensive classic, though its clarity sometimes suffers due to the authors' insistence on elementary methods (Apostol n.d.). 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:
Popular choices for a second textbook include:
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